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. 29 /2552
1 2552
1,000
99/20 10300
0-2668-7974 0-2668-7123 2530
0-2243-1129, 0-2668-7329
Web site: http:// www.onec.go.th http:// www.thaigifted.org
371.95
691
: 2552
112
ISBN 978-974-559-685-6
1. -
2. - 3.
580 8 . 34 1
. 10230
0-2943-8373-4 0-2510-7753
. .2542 ( 2) . .2545
10
28
( 2547)
(Acceleration Program)
5
6
-
( )
ตามที่พระราชบัญญัติการศึกษาแหงชาติ พ.ศ.2542 และแกไขเพิ่มเติม (ฉบับที่ 2) พ.ศ.2545มาตรา 10 วรรคสี ่ กำหนดใหการจดัการศกึษาสำหรบับคุคลซึง่มคีวามสามารถพเิศษตองจดัดวยรปูแบบที่เหมาะสม โดยคำนงึถงึความสามารถของบคุคลนัน้ และในมาตรา 28 ยงัไดกำหนดใหหลักสตูรการศกึษาสำหรับบุคคลซึ่งมีความสามารถพิเศษตองมีลักษณะหลากหลาย ทั้งนี้ ใหจัดตามความเหมาะสมของแตละระดบั โดยมงุพฒันาคณุภาพชวีติของบคุคลใหเหมาะสมแกวยัและศกัยภาพ
. . 2542 ( 2) . .2545
10 ( 4)
28
2543
School in school Program 2547
Gifted Education
(Acceleration Program)
(Enrichment Program)
( 3 6 5
6 )
(Extension Program) (mentor)
AP Program (Advanced Placement Program)
18
(Acceleration
Program) 5 ( 6 )
( )
Pre-test Post-test
1 1.
2.
3.
4.
10
6
24
38
.
4
2 5.
6.
7.
8.
38
30
48
6
200
1 9.
10.
11. 2 3
12.
27
20
36
24
5
2 13.
14.
15.
15
38
40
.
200
6
1
16.
17.
18.
(12 )
(25 )
(13 )
30
20
50
100
เรือ่ง หนาแผนการจดัการเรยีนรทูี ่1 เรือ่ง กฎเกณฑเบือ้งตนเกีย่วกบัการนบั 1
ใบความรทูี ่1.1 3ใบกจิกรรมที ่1.1 5ใบความรทูี ่1.2 8ใบกจิกรรมที ่1.2 10ใบงานที ่1 12ใบแบบฝกหดัที ่1 14
แผนการจดัการเรยีนรทูี ่2 เรือ่ง แฟกทอเรยีล (factorial ) 15ใบความรทูี ่2 17ใบงานที ่2 20ใบแบบฝกหดัที ่2 22
แผนการจดัการเรยีนรทูี ่3 เรือ่ง วธิเีรยีงสบัเปลีย่นของสิง่ของทีแ่ตกตางกนัในแนวตรง 23ใบความรทูี ่3.1 26ใบกจิกรรมที ่3.1 31ใบงานที ่3 32ใบความรทูี ่3.2 34ใบกจิกรรมที ่3.2 38ใบแบบฝกหดัที ่3 39
แผนการจดัการเรยีนรทูี ่4 เรือ่ง วธิเีรยีงสบัเปลีย่นของสิง่ของทีแ่ตกตางกนัแบบวงกลม 40ใบกจิกรรมที ่4 43ใบความรทูี ่4.1 45ใบงานที ่4 49ใบความรทูี ่4.2 51ใบแบบฝกหดัที ่4 53
แผนการจดัการเรยีนรทูี ่5 เรือ่ง วธิเีรยีงสบัเปลีย่นของสิง่ของทีม่บีางสิง่ซ้ำกนัและการแบงกลมุ 54ใบความรทูี ่5.1 57ใบงานที ่5 61ใบความรทูี ่5.2 64ใบแบบฝกหดัที ่5 67
ส า ร บ ัญส า ร บ ัญ
เรือ่ง หนาแผนการจดัการเรยีนรทูี ่6 เรือ่ง วธิจีดัหมู 69
ใบความรทูี ่6.1 72ใบงานที ่6 78ใบความรทูี ่6.2 80ใบแบบฝกหดัที ่6 82
แผนการจดัการเรยีนรทูี ่7 เรือ่ง โจทยระคนวธิเีรยีงสบัเปลีย่นและวธิจีดัหมู 84ใบความรทูี ่7 86ใบงานที ่7 90ใบแบบฝกหดัที ่7 92
แผนการจดัการเรยีนรทูี ่8 เรือ่ง ทฤษฎบีททวนิาม (binomial theorem) 93ใบความรทูี ่8 96ใบงานที ่8 100ใบแบบฝกหดัที ่8 102
⌦⌦ 1
1
6
3
*****************************************************************************************
1.
1.1
1.2 1 2
1.3 1 2
2.
3.
3.1 1
3.2 2
3.3 1 2
4.
1
4.1
4.2
4.3 1.1
4.4 1.1 1
4.5 1.1
2 - 3
4.6
2
4.7 1.2
4.8 1.2 2
4.9 1.2
⌫ ⌫ ⌦⌫ ⌫⌫2
4.10 1
4
4.11
4.12 1
5. /
5.1 1.1 1.2 (1) (2)
5.2 1.1 1.2 (1) (2)
5.3 1
5.4 1
5.5
6.
7.
7.1
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7.2
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7.3
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8.
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.…………………………………………………………………………………………………………………….
⌦⌦ 3
1.1 (1)
*********************************************************************************************
(1)
1 6 3
1
2
1 6
2 3
6 x 3 = 18 Ans.
2 2 1
2
1 1 2
2 1 2 2
2 x 2 = 4
H T
1 2 1 n1
1 2 n2
n1n2
1 2
H (H,H)
H
T (H,T)
H (T,H)
T
T (T,T)
4 (H,H) , (H,T) , (T,H) (T,T) Ans.
⌫ ⌫ ⌦⌫ ⌫⌫4
3 1
(1)
(2)
2
(1) 1 4 4
2 4
4 x 4 = 16 Ans.
(2) 1 4 4
2 3
4 x 3 = 12 Ans.
4 2 10
(1)
(2)
(3)
2 2 1 2
(1) 1 1 10
2 2 10
10 x 10 = 100 Ans.
(2) 1 1 10
2 2 9
10 x 9 = 90 Ans.
(3)
= -
= 100 - 90
= 10 Ans.
⌦⌦ 5
1.1 (1)
******************************************************************************************************
1.1.1
1. ……………
1 …………………………………………………………………………………………………...
2 ……………………………………………………………………………………………………
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2. …………………….
1 …………………………………………………………………………………………………………
2 …………………………………………………………………………………………………………
3. ( 1)
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1 …………………………………………………………………………………………………………
2 ………………………………………………………………….……………………………………..
4. 2 3
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5. 4
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6.
1 1 ( ) 2 ( ) …….
2 1 2 …… x .…..
7. 6
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2 Lee 3
A Z , Ten Co
⌫ ⌫ ⌦⌫ ⌫⌫6
1.1.2
1.
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2. 1 1
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1 2
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3.
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4. 3
2
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3
3
⌦⌦ 7
1.1.3
1. 6
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2. 1
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3. 2 , 4 , 6 , 7 9
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4. 3
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5. 14 26 1
1
5.1
5.2
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6. 1
1 n1 1
2 n2 ………...…..
⌫ ⌫ ⌦⌫ ⌫⌫8
1.2 (2)
****************************************************************************************************
(2)
1 1 3
3
1 1 6
1 6 1, 2 , 3 , 4 , 5 6
2 1 2 6
3 2 3 6
6 x 6 x 6 = 216 Ans.
2 4 5
(1)
(2) 1
4 5 4
(1) 1 1 5
2 2 5
3 3 5
4 4 5
5 x 5 x 5 x 5 = 625 Ans.
(2) 1 1 5
2 2 4
1 2 4
2 k
1 n1 1
2 n2 2 3 n3
k n1 n2 n3 . . . nk
⌦⌦ 9
3 3 3
4 4 2
5 x 4 x 3 x 2 = 120 Ans.
3 7 1
7 6 1 6
1 1 7
2 2
6
3 3 5
4 4 4
5 5 3
6 6 2
6 7 x 6 x 5 x 4 x 3 x 2 = 5,040 Ans.
4 0 , 1 , 2 , 3 3
(1)
(2)
3 3
(1) 1 0 3
2 4
3 4
3 x 4 x 4 = 48 Ans.
(2) 1 0 3
2 3
3 2
3 x 3 x 2 = 18 Ans.
⌫ ⌫ ⌦⌫ ⌫⌫10
1.2 (2)
******************************************************************************************************
1.2.1
1.
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2. 1 1 ………………………………………………
1 2 ………………………………
2 3 ………………………………
3. 2 A B 3 P , Q R 2 X Y
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4. 3 …………
2
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5. 2
……….. 3
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2 3 2
⌦⌦ 11
1.2.2
1. 1
30 ( …….. )
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2. 1 , 2 , 3 , 4 5 4
( …….. )
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3. 1 1
( …….. )
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4. COUNTRY
( …….. )
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5. 2
k 1 n1
1 2 n2
2 3 n3 k nk
k ………………………………………….
⌫ ⌫ ⌦⌫ ⌫⌫12
1
****************************************************************************************************
1.
10 8
1
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2. 3 3 6
(1) (2) (3)
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3. 0 , 1 , 2 , 3 , 4 5
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4. 4
(1)
(2)
(3)
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5. 3
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⌦⌦ 13
6.
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7. 7
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8. 4 1 5 2
8 3 3 4 6
1
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9. 100 999
(1) (3)
(2) (4) (ent.)
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⌫ ⌫ ⌦⌫ ⌫⌫14
1
****************************************************************************************************
1. 2 9
(1)
(2)
2. 20
1
3.
4. 5 3
5. 3 6
(1) (2) 2
6. 0 , 1 , 2 , 3 , 4 , 5
(1) 350 (2) 10
7. 4 “HISTORY”
(1) 4 (2) 4
(3) (4) Y
8. 5 “MATHEMATICS”
(1) 1 (2)
9. 5 3
10. 2 4 4
11. 1 5 4
12. 1 5 4
13. 2 , 3 , 5 , 6 , 7 9
(1) 400 (2) (3) 5 500
14. x y x
1 + y
1 =
9999
1
15. 1 2,000 1
⌦⌦ 15
2
( factorial ) 6
2
*****************************************************************************************
1.
1.1 n n
1.2
1.3
1.4
2.
n
n n x (n -1) x (n - 2) x ..... x 3 x 2 x 1
3.
3.1 n
3.2
4.
4.1 10
10 10 x 9 x 8 x . . . x 3 x 2
x 1
4.2 2
10 x 9 x 8 x . . . x 3 x 2 x1จากนั้นครูชี้แจงใหนักเรียนทราบวา จำนวนที่อยูในรูปลักษณะดังกลาวนี้ สามารถเขียนแทนดวยสัญลักษณ
⌫ ⌫ ⌦⌫ ⌫⌫16
4.3 n n
1
4.4 2
4.5 2
5. /
5.1 2
5.2 2
5.3 2
6.
7.
7.1
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7.2
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7.3
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8.
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⌦⌦ 17
2
******************************************************************************************************
( factorial )
n ! n n 1
5 ! = 5 x 4 x 3 x 2 x 1 = 120
3 ! = 3 x 2 x 1 = 6
1 ! = 1
(n + 2) ! = (n + 2) (n + !) n (n - 1) . . . (3)(2)(1)
(n - 3) ! = (n - 3) (n - 4) (n - 5) . . . (3)(2)(1)
2n ! = 2n (2n - 1) (2n - 2) . . . .(3)(2)(1)
n ! n 0 !
0 ! = 1
n ! = n (n - 1) (n - 2) (n - 3) ..... .3 .2 .1
= n (n - 1) !
n 1
1 ! = 1 (1 - 1) !
1 = 1 ( 0 ! )
= 0 !
0 ! = 1
1 1) 3 !6 ! = 3 x 2 x 1 x 6 x 5 x 4 x 3 x 2 x 1 = 4,320
2) 8
3
=
12 3
1 2 3 4 5 6 7 8 = 8 x 7 x 6 x 5 x 4 = 6,720
3) 12
9
=
9!
9! 10 11 12 = 12 x 11 x 10 = 1,320
n n !
n ! = n (n - 1) (n - 2) (n - 3) ..... .3 .2 .1
⌫ ⌫ ⌦⌫ ⌫⌫18
⌦⌦ 19
4 ( )n 1
(n - 2) = 720 (n + 1) !
( )n 1
(n - 2) = 720
( )( )( )( )
( ) !
n n n n
n
1 1 2
2 = 720
n + 1)(n)(n - 1) = 720
(n + 1)(n)(n - 1)
10 9 8 = 720
n = 9
(n + 1) ! = 10! Ans.
5 34
1
6
n
n
n
n
!
( ) !
( )
( ) !
n
34
1
6
n
n
n
n
!
( ) !
( )
( ) !
31
4 5 6
1
6
n n
n n n
n
n
( ) !
( )( )( ) !
( )
( ) !
34 5
n
n n( )( ) = 1
3n = (n - 4)( n - 5)
n2 - 12 n + 20 = 0
( n - 10)(n - 2) = 0
n = 10 , 2
n 6
n = 10 Ans.
6 n
n
!!
1
100 240
6 ! = 6 x 5 x 4 x 3 x 2 x 1 = 720 6 ! 240
7 ! = 7 x 6 ! 7 ! 240
n ! 240 n 6 , n N
n
n
!!
6
100 240
n
n
!
1
5 = 1! + 2! + 3! + 4! + 5! = 1 + 2 + 6 + 24 + 120 = 153
n
n
!!
1
100 240 153 Ans.
(
⌫ ⌫ ⌦⌫ ⌫⌫20
2
****************************************************************************************************
1.
(1) 5 ! + 7 ! (2) 10! - 8! (3) 11
9 2
!
! !
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2.
(1) ( ) ! ( )
!
n n n
n
2 12
(2) ( ) ( )
( ) !( ) !
n n
n n
2 4
1 3 (3)
nnnn
nn
)!1( )1( )32(
)!( )!2(2
22
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3.
(1) 9 10 11 12 (2) 150 149 148 99. . .
(3) (n - 3) (n - 4) (n - 5) (n - 6) (4) (n - 1) n (n + 1) (n + 2) (n + 3)
(5) (n2 - 81) (n2 - 64) (n2 - 49)
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⌦⌦ 21
4. n ( )
( ) !
n n n
n
27 12
3
= 15,120
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5. 8
10
10
8
! !
( ) !
!
( ) !
n
n
n
n
n
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6. 47 ! 2538
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7. 1 1! + 2 2! + 3 3! + . . . + 50 50!
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8. 1 3 5 7 9 2 1 . . . ( )n = ( ) !
( ) !
2
2
nn
n
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………………………………………………………………………………………………………………
………………………………………………………………………………………………………………
⌫ ⌫ ⌦⌫ ⌫⌫22
2
****************************************************************************************************
1.
(1) 84
(2) 711
(3) 2 86
9 6
(4) 107
3
(5) nn ( )5
(6) ( )
( )
n
n
1
1
(7) ( )
(
n
n
1 (n +1)
)2 (8)
n n
n (n +1)
( )
( )
2
1 (9) n
n k ( )
2.
(1) 100 . 99 . 98 . 97 (2) 9 . 10 . 11 . 12
(3) 20 . 19. 18 . ..... . 10 (4) (n+2) (n+1) n (n-1)
(5) n (n-1)(n-2) ..... (n-9) (6) n (n-1)(n-2) ..... (n - r + 1)
(7) n (n+1)(n+2) ..... (n + r ) (8) 3n ( 9n2 - 9n + 2 )
3. n
(1) ( )
( )
n
n
3
1
= 1680 (2) 2 .
( )
( )
n
n
4
2 =
( )
( )
n
n
5
1
(3) ( )n
n
1
= 1
2 .
( )
( )
n
n
2
1
(4) 3 .
( )
( )
2 4
2 1
n
n = 2 .
( )n
n
4
(5) nn
( )4
= 30 . nn
5 ( )5
⌦⌦ 23
3
6
4
*****************************************************************************************
1.
1.1
1.2 n
1.3 n
r
2.
2.1
2.2 n n
n!
2.3 n r
P(n , r) = n
n r
!
( ) !
3.
3.1 ( 3)
3.2 ( 4)
4.
1 - 2
4.1
⌫ ⌫ ⌦⌫ ⌫⌫24
4.2 3.1
4.3
( 3)
4.4 3.1
4.5
4.6 3
3 - 4
4.7 3.2
4.8
( 4)
4.9 3.2
4.10
4.11 3
5. /
5.1 3.1
5.2 3.1
5.3 3
5.4 3.2
5.5 3.2
5.6 3
6.
⌦⌦ 25
7.
7.1
……………………………………………………………………………………………………………..………
…………………………………………………………………………………………………...………………...
…………………………………………………………………………………………..........................................
....……………………………………………………………………………………………………………..……
7.2
……………………………………………………………………………………………………………..………
…………………………………………………………………………………………………...………………...
…………………………………………………………………………………………..........................................
....……………………………………………………………………………………………………………..……
7.3
……………………………………………………………………………………………………………..………
…………………………………………………………………………………………………...………………...
…………………………………………………………………………………………..........................................
....……………………………………………………………………………………………………………..……
8.
……………………………………………………………………………………………………………..………
…………………………………………………………………………………………………...………………...
…………………………………………………………………………………………..........................................
……………………………………………………………………………………………………………..………
⌫ ⌫ ⌦⌫ ⌫⌫26
3.1
****************************************************************************************************
( permutation )
5
3 3
3
6 2 3
7
2
1.
2.
1. 5 5
5 x 4 x 3 x 2 x 1 5 !
2. EQUATION 8
8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 8 !
1 7
7 7
7! = 5,040 Ans.
3 n
n !
⌦⌦ 27
2 “SECOND”
6 4 2
6
2 2
1
4 4!
2 x 1 x 4! = 48 Ans.
3 9 3
(1) 3 (2) 3
(3) 3
9 9
(1) 3 3
1
1 2 3
9 7
2
1 7 7!
2 3 3!
7! x 3! = 5,040 x 6 = 30,240 Ans.
(2) 3 (1)
3
3
= 9!
3 = 9! - 7! 3! = 332,640 Ans.
⌫ ⌫ ⌦⌫ ⌫⌫28
(3) (2) 3 2
6
2
1 6 6!
2 3 7 7 x 6 x 5 = 210
= 6! x 210 = 151,200 Ans.
4 6 6
(1) 1 (2) 2 (3) 3
(1) 2
1
2
2
1 12 12
= 6 x 6 x 5 x 5 x 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1
= 6! 6!
2 6! 6! + 6! 6! = 2 (6! 6!)
2 2
6 6 = 6!
6 6 = 6!
= 6! 6!
2 6! 6! + 6! 6! = 2 (6! 6!) Ans.
⌦⌦ 29
(2) (1) 2 2
6 6 6!
6 6 6!
= 6! 6!
2 6! 6! + 6! 6! = 2 (6! 6!) Ans.
(3) (1) (2) 3 2
6 6 6!
6 6 6!
= 6! 6!
2 6! 6! + 6! 6! = 2 (6! 6!) Ans.
n
k k n
2 x n ! x n !
5 6 2 3
1 2 3
4 5 6
6
6! = 720 Ans.
⌫ ⌫ ⌦⌫ ⌫⌫30
6 6 12
2 6
(1) (2)
(1)
2
6 6!
6!
2 (6! 6!) Ans.
(2)
⌦⌦ 31
3.1
****************************************************************************************************
1. 3 3 ………………………………………
2. 1 ……………………………………………..…………
3. 10 …………………………………………..
4. 3 ……………………………………………..…………
5. 9 ……………………..
6. 5 …………………… 1, 2, 3, 4, 5
………………
7. 7 …………………………………………………..……
8. 7 ……………………………
( ) ………………………………… 7
10. 6 6
……………………………………………………
11. 10 ( ) ……………………………………….
………………….………………………………..………………………………
12. 10 1 ………..…......
( ) ……………………. 10
13.
( ) …………………………………………….………………
14. 13 ( ) ………………
15.
“
” ……………………
16. 2
………………………………..….
17. 16 n n
……………………………………… ( 3 )
*********************************************
⌫ ⌫ ⌦⌫ ⌫⌫32
3
****************************************************************************************************
1. “NUMBER”
(1) (2)
(3) (4)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
2. 7
7
(1) (2)
(3) (4)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
⌦⌦ 33
3. 5 4 9
(1) 2 (2) 2
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
4. 1 , 2 , 3 , ..... , 9 1
1 , 2 3
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
5. 4 1 , 2 , 3 4
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
(1)
(2)
⌫ ⌫ ⌦⌫ ⌫⌫34
3.2
*****************************************************************************************************
1. 8 5 5
8 x 7 x 6 x 5 x 4
8 3
= 8 (8 -5)
2. 4 2 , 4 , 5 , 7 , 8 , 9
4
6 x 5 x 4 x 3
6
2
! =
6
6 4
( ) !
n r
P(n , r)
1) r = n
P(n , n) = n (n - n)
= n0
= n !
n
2) nPr nPr Pn,r P(n , r)
4 n r
n(n - r)
( r n )
P(n , r) = n(n - r)
⌦⌦ 35
1 P(4 , 2) = 4
4 2
4
2
!
( ) !
!
! = 4 x 3 = 12
P(9 , 3) = 9
9 3
9
6
!
( ) !
!
! = 9 x 8 x 7 = 504
P(15 , 5) = 15
15 5
15
10
!
( ) !
!
!
= 15 x 14 x 13 x 12 x 11 = 360,360 Ans.
2 n 3. P(n , 4) = P(n-1 , 5)
3. P(n , 4) = P(n-1 , 5)
34
1
1 5
34
1
6
31
4 5 6
1
6
3
4 51
n
n
n
n
n
n
n
n
n n
n n n
n
n
n
n n
!
( ) !
( ) !
( ) !
!
( ) !
( ) !
( ) !
( ) !
( )( )( ) !
( !
( ) !
( )( )
n2 - 12n + 20 = 0
( n - 10 )( n - 2 ) = 0
n = 10 , 2
n = 2
n = 10 Ans.
3 10 3
10 3
P(10 , 3) = 10
10 3
10
7
!
( ) !
!
!
= 10(9)(8)
= 720 Ans.
4 5 3 1 2
3
5 3
P(5 , 3) = 5
5 3
5
2
!
( ) !
!
!
= 5(4)(3)
= 60 Ans.
⌫ ⌫ ⌦⌫ ⌫⌫36
5 5 “PERMUTATION”
(1) (2)
(3) (4) 1
“PERMUTATION” 10 5
5
5 10
10 5
(1) P(10 , 5) = 10
10 5
10
5
!
( ) !
!
!
= 10(9)(8)(7)(6)
= 30,240 Ans.
(2) 5
1 5
2 4
9
4 P(9 , 4)
5 x P(9 , 4) = 5 x 9
5
!
!
= 5(9)(8)(7)(6) = 15,120 Ans.
(3) 5
1 5
2 4
9
4 P(9 , 4)
5 x P(9 , 4) = 5 x 9
5
!
!
= 5(9)(8)(7)(6) = 15,120 Ans.
(4)
1
( )
= 5! = 120
1 = 30,240 - 120 = 30,120 Ans.
⌦⌦ 37
6 5
1
5 1 2 3
4 5
1 1 5 5
2 2 P(5 , 2) = 20
2 5
3 3 P(5 , 3) = 60
3 5
4 4 P(5 , 4) = 120
4 5
5 5 5! = 120
5
5 + 20 + 60 + 120 + 120 = 325 Ans.
7 1 , 2 , 3 , 4 , 5 , 6 4
6 4
4 = P(6 , 4) = 6
6 4
6
2
!
( )!
!
!
= 360
360
1 , 2 , 3 , 4 , 5 6 360
6 = 60
(1 x 60) + (2 x 60) + (3 x 60) + (4 x 60) + (5 x 60) + (6 x 60) = 1,260
1,260
10 , 100 1,000
1,260 + (1,260 x 10) + (1,260 x 100) + (1,260 x 1,000) = 1,399,860 Ans.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
⌫ ⌫ ⌦⌫ ⌫⌫38
3.2
******************************************************************************************************
1. 5 3 ………………………………………
5 ………………………….
2. 1 ………………………………………………………..
3. SOCIAL 4 ………………
6 ……………………………….
4. 3 …………………………………………………..……
5. 9 6 ……………………
9 …………………………………….
6. 5 ………………………………………………………..
7. 1, 2, 3, 4, 5 ………………
5 …………………………………….
8. 7 ………………………………………………………..
9. 7 ……………………………………
( ) ………………………………………. 7
10. 5 8
………………………………………
8 …………………………………….
11. 10 ( ) …………………….…………………
………………………………………………………………………………......
12. 10 1 ……………..
( ) ……………………. 10
13. ( ) …………………………
14. n r
……………………………………………. ( 4 )
15. n r
P(n , r) P(n , r) ……………………………………………………
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
⌦⌦ 39
3
******************************************************************************************************
1. 7 A , B , C , D , E , F G
2. 22
3. 4 2
4. 3 4 2
5. 5
6. 12 12
(1) (2) 1
(3) 2 (4) 3
7. 6 6
(1) (2)
(3)
8. 4 3 3
(1) (2) 3
(3) 3
9. 8 4
(1) 4 (2) 3
(3) 2 (4) 3
10. 8 3
2 4
(1) (2)
(3)
11. 6
(1) (2) 2
(3)
12.
(1) 6 (2) 7
(3) 8
(4) 7
13. A = {1, 2, 3, 4, 5, 6 } B = { {1} , {1, 2}, {1, 2, 3 } } f : B A f (x) x
x B
14. 4,000,000 250
15. A = { 1, 2 , 3, 4, 5, 6 } B = { 1, 2, 3 } f : A B
⌫ ⌫ ⌦⌫ ⌫⌫40
4
6
3
*****************************************************************************************
1.
1.1 2
1.2 2 n
1.3 2 n
r
1.4 3
1.5 3 n
1.6 3 n
r
2.
2.1
2.2 2 2
3
2.3 n 2 n
= ( n - 1)!
2.4 n 2 r
= P n r
r
n
r n r
( , ) !
( ) !
2.5 3
2
3.
3.1
3.2 2 ( 5 6)
3.3 3 ( 7 8)
⌦⌦ 41
4.
1- 2
4.1 5
5 3
4.2 5
4.3 2
4.4 3
4.5 3
4.6 4
4.7
3
4.8
4.9
4.10 4
4
4.11 4 5 6
2 4
4.12 4.1
2 4
3
4.13 4
4.14 4.2
4.15 4.2 ( )
⌫ ⌫ ⌦⌫ ⌫⌫42
4.16 7 8
3 4.2
4.17
3 4
5. /
5.1 4 2
5.2 4.1 2
5.3 4 2
5.4 4.2 3
5.5 4
5.6
5.7
6.
7.
7.1
…………………………………………………………………………………………………………………......
……………………………………………………………………………………………………………………..
……………………………………………………………………………………………………………………..
7.2
…………………………………………………………………………………………………………………......
……………………………………………………………………………………………………………………..
……………………………………………………………………………………………………………………..
7.3
…………………………………………………………………………………………………………………......
……………………………………………………………………………………………………………………..
……………………………………………………………………………………………………………………..
8.
…………………………………………………………………………………………………………………......
……………………………………………………………………………………………………………………..
……………………………………………………………………………………………………………………..
⌦⌦ 43
4 2
****************************************************************************************************
4.1
n
1. n = 2 …………………………………
………………………………………………………………..……
2. n = 3 …………………………………
……………………………………………………………………..
3. n = 4 …………………………………
……………………………………………………………………..
4.2
1. 2 A , B ………… ……………………………….
AB BA 2 …………………………….
2. 3 A , B , C ………………
……………………………………………………………………………………………………………
ABC BCA CAB 3 ………………………
ACB CBA BAC 3 ………………………
3. 4 A , B , C , D ………………
ABCD BCDA CDAB DABC 4 ……………………
ACBD CBDA BDAC DACB 4 …………..……….
4. 5 A , B , C , D , E ………………
ABCDE BCDEA CDEAB DEABC EABCD 5 …….…
5.
5.1 2 ; 2 1
2! ……………
………………………………………………………………………
5.2 3 ; 3 ……………
3! ……..…
………………………………………………………………………
⌫ ⌫ ⌦⌫ ⌫⌫44
5.3 4 ; 4 ………...…
4! ……...…
………………………………………………………………………
5.4 5 ; 5 …………...
5! ……….
………………………………………………………………………
5.5 n ; n ………..…
n! …………
………………………………………………………………………
6. n 2
……………………………….. ( 5 )
7. 3 A , B C 2 …………………
……………………………………………………………………………………………………………
2 …………………..
P(3 , 2) …………………..
8. 4 A , B , C D 3 …………
3 …………………..
P(4 , 3) …………………..
9. n r ………………
r …………………..
P(n , r) …………………..
10. n 2
r ……………………………….. ( 6 )
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
⌦⌦ 45
4.1 2
******************************************************************************************************
A , B , C 3 ! = 6
ABC BCA CAB
ACB CBA BAC
2
ABC BCA CAB
ACB CBA BAC
2
1. 2
2. 3
2
A , B , C
3 3 1
3 3! 33 ! = 2!
A
BC
A
B C
⌫ ⌫ ⌦⌫ ⌫⌫46
4 A , B , C , D 4 ! = 24
4
ABCD BCDA CDAB DABC
4 4 1
4 4 ! 44 ! = 3 !
2 n
n - 1 (n - 1 ) !
n r
n r P(n , r)
r 1
P(n , r) P n r
r
( , ) = n
n r! ! r( )
A
B
C
D
5 n
2 (n - 1 ) !
6 n
2 r P n r
r
( , )
⌦⌦ 47
1 6 4 2
(1) (2) 2
(3) 2 (4) 2
(1) 2
6 (6 - 1)! = 5! = 120 Ans.
(2) 2 2 1
6 5
5 (5 - 1)! = 4!
2 2!
4! x 2! = 24 x 2 = 48 Ans.
(3) 2
2
2
120 - 48 = 72 Ans.
(4)
n n - 1
5
1 1
4 4!
1 x 4! = 24 Ans.
2 8 ,
(1) (2) , ,
(3) , , (4)
(1) 8 8 2
(8 - 1)! = 7! = 5,040 Ans.
(2) , 1
8 6
6 5!
, 3!
5! x 3! = 120 x 6 = 720 Ans.
⌫ ⌫ ⌦⌫ ⌫⌫48
(3) , , 3
, ,
5 4!
, , 5 P(5 ,3)
4! x P(5 ,3) = 24 x 5 x 4 x 3 = 1,440 Ans.
(4) 7
2
5
5 5!
2 x 5 x 5! = 1,200 Ans.
3 10 5
10 5
p( , ) !
( !)
10 5
5
10
5 5
= 2(9)(8)(7)(6) = 6,048 Ans.
#####################################
⌦⌦ 49
4 2
******************************************************************************************************
1. 4 2 2
2 3
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
2. 4 2
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
3. 6 6
1) 1 2) 2 3) 3
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
⌫ ⌫ ⌦⌫ ⌫⌫50
4. 1 3 3
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
5. 7
(1)
(2)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
6. 15 10
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
***********************************************
⌦⌦ 51
4.2 3
******************************************************************************************************
3
3 2
2 A , B , C , D 3 ! = 6
3
1 2
3 4
5 6
3 3
2
A
B
C
D
1.
A
B
C
D
2.
A
B C
D
4.
A
B
C D
6.
A
BC
D
3.
A
B
CD
5.
7 n 3
( )n 1
2
!
⌫ ⌫ ⌦⌫ ⌫⌫52
1 7
3
7 7
= ( ) ! !7 1
2
6
2
720
2 = 360 Ans.
2 9 2 3
(1) (2)
3
(1) 9 9
= ( ) ! ! ,9 1
2
8
2
40 320
2
= 20,160 Ans.
(2)
1 1
9 6
6 = ( ) ! !6 1
2
5
2
120
2 = 60
2 2! = 2
3 3! = 6
= 60 x 2 x 6 = 720 Ans.
3 10 6
3
10 6
= P( , )
( )
!
( )( !)
10 6
2 6
10
2 6 4 = 12,600 Ans.
@@@@@@@@@@@@@@@@@@@@@@@@@
8 n 3 r
P n r
r
( , )
2
⌦⌦ 53
4
******************************************************************************************************
1. 10
(1)
(2)
2. 4 3
(1) (2)
3. 5
(1) (2) 1
4. 5 1
(1)
(2)
5. 7 ,
6. 12 12
(1) (2) 1
(3) 2 (4) 3
7. 6 5
(1) 2 1
(2) 2 1
8. 4 3 2
(1) (2)
9. 8
(1) (2) (3)
10. 10
5
⌫ ⌫ ⌦⌫ ⌫⌫54
5
6
5
*****************************************************************************************
1.
1.1
1.2
1.3
1.4
1.5
2.
2.1 n k 1 n1 2
n2 . . . k nk n1 + n2 + ..... +nk = n
n n !n1! n
2n
k ! ..... !
2.2 n k 1 n1
2 n2 ........ k nk n1 + n2 + ..... + nk = n
n !n1! n
2n
k ! ..... !
2.3 n k 1 n1 2 n2
............. k nk n1 + n2 + ..... + nk = n . . (n1 , n2 , ..... , nk )
= 1 n ( )n 1 !
n1! n
2n
k ! ..... !
⌦⌦ 55
3.
3.1 ( 9)
3.2
3.3 ( 10)
4.
1
4.1 A , B , C
4.2 A B M
4.3 3 M , M , C
4.4 “ELEMENT” E
3
4.5
( 9)
2
4.6
5.1
3
4.7
5
4
4.8 5
4.9 5.2
( 10)
5
4.10
5
⌫ ⌫ ⌦⌫ ⌫⌫56
5. /
5.1 5.1
5.2 5
5.3 5.2
5.4 5
6.
7.
7.1
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.2
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.3
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
8.
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
⌦⌦ 57
5.1
******************************************************************************************************
1. 3 A , B , C 3 ! = 6
ABC ACB BAC BCA CAB CBA
A B M 6
MMC MCM MMC MCM CMM CMM
3 MMC MCM CMM
M , M , C 3
3 MMC M 2
2 !
1 (MMC) 2 !
3 3 x 2 ! = 3 ! = 6
3 3 !2 !
2. ”ELEMENT”
7 E 3
= y
E 3 !
y E 3 ! x y
E 7 = 7 !
3 ! x y = 7 !
y = 7 !3 !
“ELEMENT” 7 !3 !
3 3 !2 !
⌫ ⌫ ⌦⌫ ⌫⌫58
1 500 3 100 4 20 2 10
3 (
)
12
500 3
100 4
20 2
10 3
= 12
3 4 2 3
!
! ! ! !
= 277,200 Ans.
2 4 4 3
(1) (2)
(3) 3
(1) 11 4
= 11
4
!
!
= 1,663,200 Ans.
9 n k
1 n1
2 n2
k nk
n1 + n2 + ..... +nk = n
n n !n1! n
2n
k ! ..... !
…..
…..
⌦⌦ 59
(2) 11 3
3 3!
4 4!
3 3!
4
= 3! 4! 3! = 864 Ans.
(3)
4 4 8
4
!
! = 1,680
8 3 9 P(9 , 3) = 9
6
!
! = 504
1,680 x 504 = 846,720 Ans.
3 7 1 , 1 , 2 , 2 , 2 , 3 , 4 7
(1) (2) 2
(1) 7 1 2 2 3
7 7 7
2 3
!
! ! = 420 Ans.
(2) 2 3 4 2
1 2 1
2 2 1 2
6 6
2 2
!
! ! = 180
7 1 x 180 = 180 ***
2 3 4 2
2 3 1 2
6 6
3 2
!
! ! = 60
7 2 x 60 = 120 ***
7 2 180 + 120 = 300
Ans.
9 “ ” ( partitioning law )
n k 1 n1
2 n2 ........ k nk n1 + n2 + ..... + nk = n
n !n1! n
2n
k ! ..... !
⌫ ⌫ ⌦⌫ ⌫⌫60
n
k m
n
m kk
!
( !) !
4 10 2
3 5
(1)
(2) 1 10
5
(1) 10 3 2 , 3 5
10
2 3 5
!
! ! ! = 2,520 Ans.
(2) 9 3 2 ,3 4
9
2 3 4
!
! ! ! = 1,260 Ans.
5 10 2 2
5
2
10
5 5
!
! ! = 252 Ans.
* 6* 10 2
5
10
5 5 2
!
! ! ! = 126 Ans.
* 7* 9 3 4 3
2 3 4
1 9 3 2 , 3 4 9
2 3 4
!
! ! ! = 1,260
2 3 3 3! = 6
1,260 x 6 = 7,560 Ans.
@@@@@@@@@@@@@@@@@@@@@@@
⌦⌦ 61
5
******************************************************************************************************
1. “MISSISSIPPI”
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
2. 3 2 4
(1) (2) 2
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
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……………………………………………………………………………………………………………………
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3. 9 2,520
……………………………………………………………………………………………………………………
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……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
⌫ ⌫ ⌦⌫ ⌫⌫62
4.
A B
(1)
(2) C
(3) C D
(4) C D
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
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……………………………………………………………………………………………………………………
5. 9 3 4 3 2
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
A
.
.
C.
.
B
D
⌦⌦ 63
6. 10 3
(1) 4 2 3
(2) 4 2 3
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
7. 10
3 4 3 3
10
(1) (2)
(3)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
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8. 6 3 3
……………………………………………………………………………………………………………………
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……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
⌫ ⌫ ⌦⌫ ⌫⌫64
5.2
******************************************************************************************************
: 1. 10 n1 , n2 , ..... , nk 1 . .
(n1 , n2 , ..... , nk ) = 1
2. . . (n1 , n2 , ..... , nk ) 1 (
) AABBCC
( . . . 2 , 2 , 2 2) 16
( ) !
! ! !
6 1
2 2 2 = 15
(Mobius Theorem)
1 2 3 4
9 2 3
4 . . . 2 , 3 4 1
( ) !
! ! !
9 1
2 3 4 = 140 Ans.
10 n k 1 n1
2 n2 ............. k nk
n1 + n2 + ..... + nk = n . . (n1 , n2 , ..... , nk ) = 1
n ( )n 1 !
n1! n
2n
k ! ..... !
⌦⌦ 65
2 1 2 2
3
(1) (2) (3) 3
(1) 8 2 2
3 1
. . . 2 , 2 , 3 1 1
( ) !
! ! !
8 1
2 2 3 = 210 Ans.
(2) 4
3! = 6 Ans.
(3) 1 6 6
2 2 1 1
. . . 2 , 2 , 1 1 1
( ) !
! !
6 1
2 2 = 30 Ans.
3 8 2
2
B W G
B 2 W 2
G 4 B
B 7
BB WWGGGG
. . . 1 , 2 4 1
( ) !
! !
7 1
2 4 = 15
15 Ans.
BB WWGGGG
⌫ ⌫ ⌦⌫ ⌫⌫66
4 3 8 6
6 2
B 6 W 2
. . . 6 2 2
4 Ans.
5
1 2 2
3 = 3
2 2 1
6 5
. . . 1 , 2 2 1
= ( ) !
! !
5 1
2 2 = 6
= 3 x 6 = 18 Ans.
r n
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
แบบที ่1W ตดิกนั 2 ชอง
แบบที ่2W 1 ชอง เวน 1 ชอง
แบบที ่3W 1 ชอง เวน 2 ชอง
แบบที ่4W 1 ชอง เวน 3 ชอง
W W
B BBB
BB
W B
B BBB
BW
W B
B BBB
WB
W B
B WBB
BB
⌦⌦ 67
5
******************************************************************************************************
1. “COOPERATOR”
(1) (2) O 3
(3) R 2
2. “MATHEMATICIAN”
(1) A 3 (2) A 2
3. “TROTTING”
T
4. 4 3
(1) (2) (3)
5. 7 2 2 3
7
(1)
(2)
6. 2 3 2
7 1
7. 9 3 4 3 2
8. 8 3 4 2 2
9. 4 4 2 2
10. 9 3 3
11. 4 4 2
12. A B
A
B
⌫ ⌫ ⌦⌫ ⌫⌫68
13. 11
11 5
14. 8
(1) 4 4
(2) 2 3 3
(3) 6 1 1
15. 9 3
3 3
(1) (2) 3
(3) 1 (4) 1
16. 10
12,600
(1)
(2)
17. 5 3
18. “PARALLEL”
(1)
(2) A
19. X, X, Y, Y, Y, Z, Z
(1) (2) X 2
(3) Y 3 (4) Z
20. 2 4 3
3
⌦⌦ 69
6
6
4
*****************************************************************************************
1.
1.1
1.2
1.3 r n
1.4 r n
2.
2.1 r n
2.2 n r n
r n r
!
! ( ) !
3.
3.1 r n ( 11)
3.2 r
4.
1 - 2
4.1 A , B , C , D 3 4
ABC , ABD , ACD , BCD
4.2 3
4 A , B , C , D
4.3 3 4
4.4
4.5 -
⌫ ⌫ ⌦⌫ ⌫⌫70
1) 3 4
( 3 ! )
2) 4 ( 3 ! x 4 )
3) 3
4 [ P(4 , 3) ]
4 x 3! = P(4 , 3)
4 = !3
)3,4(P
3 4 = !3
)3,4(P
4.6
r n ( 11)
4.7 C (n , r) C (n , r) n r
4.8 -
4.9 6.1
3
4.10 6.1
4.11 r
n 6
4.12 6
4
4.13
11
4.14 1 ( )
4.15
4.16 r
n
4.17 6.2
⌦⌦ 71
4.18 r
n 6
5. /
5.1 6.1 r n
5.2 6 r n
5.3 6.2 r n
5.4 ( )
5.5 6
6.
7.
7.1
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.2
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.3
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
8.
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
⌫ ⌫ ⌦⌫ ⌫⌫72
6.1 r n
******************************************************************************************************
( combination )
( )
r n
r n
4 A , B , C , D 3
ABC , ABD , ACD BCD 4
3! 3 4 4 x 3!
3 4 P(4 , 3)
4 x 3! = P(4 , 3)
4 = !3
3) , P(4
3 4 = !3
3) , P(4
r n = !r
r) ,P(n
=
r n nCr nCr
n,r C (n , r) n
r
n
r
11
11 n r
nr n r
! !! ( )
n
r = nr n r
! !! ( )
⌦⌦ 73
1
6
0
6
0 6 0
6
0 61
6
1
6
1 6 1
6
1 56
6
2
6
2 6 2
6
2 415
6
3
6
3 6 3
6
3 320
6
4
6
4 6 4
6
4 215
6
5
!
!( ) !
!
! !
!
!( ) !
!
! !
!
!( ) !
!
! !
!
!( ) !
!
! !
!
!( ) !
!
! !
6
5 6 5
6
5 16
6
6
6
6 6 6
6
6 01
!
!( ) !
!
! !
!
!( ) !
!
! !
16
0
6
6
6
1
6
5
6
2
6
4
26
0
6
1
6
2
6
3
6
4
6
5
6
664 2
6
. , ,
.
12 10
10 . . .
3
10
2
10
1
10 .4
2 8
8 . . .
2
8
1
8
0
8 .3
2 4
4
3
4
2
4
1
4
0
4 .2
3
5
2
5 ,
4
5
1
5 ,
5
5
0
5 .1
10
8
4
1
20 1 2 3
2
.
. . . .
n
r
n
n r
n n n n n
n
n
⌫ ⌫ ⌦⌫ ⌫⌫74
2 10
(1)
(2)
10 3
(1) 1 2
2 10
10
2
10
2 8
!
! ! = 45 Ans.
(2) 1 3
3 10
10
3
10
3 7
!
! ! = 120 Ans.
3 12
(1) 2 12
(2) 12 A , B , C D
4 3 2 12
(1) 12 3
2 12
2
12
2 10
!
! ! = 66 Ans.
(2) 12 4
2 4
4 2
4
2
4
2 2
!
! ! = 6
4 6 1
2 12
2 - 6 + 1 = 61 Ans.
⌦⌦ 75
4 2 6 7 2
1 1
2
1
1
1 6
2
6
2 4
!
! ! = 15
1 7
2
7
2 5
!
! ! = 21
15 x 21 = 315 Ans.
5 5 1
(1) (2) 4
(3) 2 3 (4) 5
1 52 4 ( ) ( )
( ) ( ) 13
A , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , J , Q K
(1) 5 52
52
5
52
5 47
!
! ! = 2,598,960 Ans.
(2) 1 4 4 4
4 = 1
1 48
1 = 48
1 x 48 = 48 Ans.
(3) 1 13 13
2 13
2
13
2 11
!
! ! = 78
3 13
3
13
3 10
!
! ! = 286
2 3 78 x 286 = 22,308 Ans.
⌫ ⌫ ⌦⌫ ⌫⌫76
(4)
1 4 4
1 = 4
5 13 13
5
13
5 8
!
! ! = 1,287
5 4 x 1,287 = 5,148 Ans.
6 9 6
(1) 3
(2)
(1) 3 9
3 = 84
3 6
3 = 20
3 84 x 20 = 1,680 Ans.
(2) 1 , 2 , 3 , 4 , 5 6
9
1
6
1
9
2
6
2
9
3
6
3
9
4
6
4
9
5
6
5
9
6
6
6 = 5,004
5,004 Ans.
7 5 2 1 1
4 10
(1) 2 2 (2)
(3) 1
(4) 2 2 1
(1) 2 5 5
2 5 5
2 = 10
2 5 5
2 = 10
2 2 10 x 10 = 100 Ans.
⌦⌦ 77
(2) 4 4 4
5 4 5 5
4 = 5
2 1 2
1
2
1
2
1
2
1 = 16
5 x 16 = 80 Ans.
(3) 1 1 5
5
1 = 5
2 2 2 2 4
4
2 = 6
2 1 2
1
2
1 = 4
1 5 x 6 x 4 = 120 Ans.
(4) 1
1 2
1 1
1 5
1 = 5
1 4 4
1 = 4
1 3 3
1 = 3
5 x 4 x 3 = 60 ***
2 2
2 5 5
2 = 10 ***
2 60 + 10 = 70 Ans.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
⌫ ⌫ ⌦⌫ ⌫⌫78
6 r n
******************************************************************************************************
1. C (n , r) = C (n , n - r)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
2. r C (n , r) = P (n , r)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
3. a b b 17
19
14 5
a
20
17
20
b
a
b
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
4.
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
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……………………………………………………………………………………………………………………
⌦⌦ 79
5. .6 9
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
6. 1 1
(1)
(2)
……………………………………………………………………………………
……………………………………………………………………………………
……………………………………………………………………………………
…………………………………………………………………………………………………………………….
….…………………………………………………………………………………………………………………
…….………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……….……………………………………………………………………………………………………………
7. A = { a , b , c , d } ; B = { p , q , r , s , t } C B C A
C
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
*******************************************
⌫ ⌫ ⌦⌫ ⌫⌫80
6.2 r n
******************************************************************************************************
r n
r n
1 6 1 1 1 3
4
4
1 1
1 1
1 1
1 1
1 1
1 x 1 x 1 x 1 = 1
2 2 2 2
2 1
2 3 3
2 = 3
1 x 3 = 3
3 3 3 1
3 1
1 3 3
1 = 3
1 x 3 = 3
1 + 3 + 3 = 7 Ans.
⌦⌦ 81
2 a , a , a , b , b , c , c , d , d , e , f 4
4
1 4
6 4 6
4 = 15
2 2 2
2 4
1 = 4
2 5
2 = 10
4 x 10 = 40
3 2 2
2 2 4
2 = 6
6
4 3 1
3 1
1 5
1 = 5
1 x 5 = 5
15 + 40 + 6 + 5 = 66 Ans.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
⌫ ⌫ ⌦⌫ ⌫⌫82
6
******************************************************************************************************
1. n
1) 4 . C (n , 2) = C (n+2 , 3) 2) C (n , 12) = C (n , 8)
3) C (n-2 , 2) + C (n-3 , 2) + C (n-4 , 2) = 10
2. P(n , r) = 3,024 C (n , r) = 126
1) C (10 , r+1) 2) C (9 , r -1)
3. C (18 , r) = C (18 , r+2) C (r , 5) P(r+1 , 3)
4. 12 8
1) 3
2) 2 3 1 2
5. 5 10 2
4 5
1) 3 1 1
2) 3 3) 3
6. 3 20
1)
2) 20 2
3) 2
7. 4
1) 2)
3) 1
4)
8. 10 6
1) 2 2) 3
3) 2
9. 10 A , B C
1) 2) A
3) A B 4) A B C
10. 10 4
⌦⌦ 83
11. 1
12.
36
13. 20
9 5 6
3 2
14.
15. 6 4
1) 4 2) 4 1
3) 4 1
16. 1
1) 4 2) 3
3) 3 4) 4 2
17. A = { 1 , 2 , 3 , 4 , 5 } B = { a , b , c , d } 1 - 1 C B C A
C
18. 2 2 3 3 3
4 2
2
1) 4 2) 2 3) 2
19. 8 6 2
4
1) 4 2) 4
3) 4
20. 1
1
1)
2)
3)
⌫ ⌫ ⌦⌫ ⌫⌫84
7
6
5
*****************************************************************************************
1.
1.1
1.2
2.
r n
3.
4.
1 - 2
4.1
1 “ 6 6
4 4 4
”
( 20 )
4.2 7
( 40 )
4.3 7
(60 )
⌦⌦ 85
3 – 5
4.4
7
6
4.5 1
4.6 7
5. /
5.1 7
5.2 7
5.3 7
6.
7.
7.1
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.2
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.3
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
8.
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
⌫ ⌫ ⌦⌫ ⌫⌫86
7
******************************************************************************************************
1 8 9
4 3 7
1 4 8 8
4
2 3 9 9
3
3 7 7 7!
= 8
4 x
9
3 x 7! Ans.
2 5 5
5 2 2
1 2 5 5
2
2 2 5 5
2
3 2 2 4!
5
2 x
5
2 x 4! = 10 x 10 x 24
= 2,400 Ans.
3 10 6 4 3
2 2
1 3 6 6
3
2 2 4 4
2
⌦⌦ 87
3 3 2 2
- 2
5! - 4! 2! = 120 - 48 = 72
= 6
3 x
4
2 x 72
= 20 x 6 x 72
= 8,640 Ans.
4 10 5 5
6
6
3 3
1 3 5 5
3
2 3 5 5
3
3 2! 3! = 12
= 5
3 x
5
3 x 12
= 10 x 10 x 12
= 1,200 Ans.
5 1 3
1 1 7 5
1 3 5 5
3
2 3 7 7
3
3 3 3 1 1
3!
= 5
3 x
7
3 x 3!
= 10 x 35 x 6
= 2,100 Ans.
⌫ ⌫ ⌦⌫ ⌫⌫88
6 5 “INTERMEDIATE”
“INTERMEDIATE” I 2 T 2 E 3 5
1 5
1 5 8 ( I , N , T , E , R , M , D , A ) 8
5
2 5 5!
= 8
5 x 5!
= 56 x 120 = 6,720
2 5 2
1 2 1 3 ( I , T , E) 3
1
2 3 7 ( 5 1
2 7 ) 7
3
3 5 2 5
2
!
!
= 3
1 x
7
3 x
5
2
!
!
= 3 x 35 x 60 = 6,300
3 5 2 2
1 2 2 3 ( I , T , E) 3
2
2 1 6 ( 5 2
1 6 ) 6
1
3 5 2 2 5
2 2
!
! !
= 3
2 x
6
1 x
5
2 2
!
! !
= 3 x 6 x 30 = 540
⌦⌦ 89
4 5 2 1 3 1
1 2 1 2 ( I , T) 2
1
2 3 1 (E) 1
3 5 2 1 3 1
5
2 3
!
! !
= 2
1 x 1 x
5
2 3
!
! !
= 2 x 1 x 10 = 20
5 5 3
1 3 1 (E) 1
2 2 7 ( 5 2 7
7
2
3 5 3 5
3
!
!
= 1 x 7
2 x
5
3
!
!
= 1 x 21 x 20
= 420
5 = 6,720 + 6,300 + 540 + 20 + 420
= 14,000 Ans.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
⌫ ⌫ ⌦⌫ ⌫⌫90
7
******************************************************************************************************
1. 8 4 5
(1) 3 2 (2) 1
(3)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
2. 5 6
(1)
(2) 3
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
3. 10
8 8 8
3 2
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
⌦⌦ 91
4. 312 259
8
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
5. 5 5
5 3 2
(1)
(2)
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
6. 6 6 1 4 4
2 4
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
*****************************************
⌫ ⌫ ⌦⌫ ⌫⌫92
7
******************************************************************************************************
1. 10 3 3
2. 3 2 7
4
1) 2)
3. 6 4 8
4. 10 5 5 6
5. 20 15
6. 5 25 20
7. 7 4
8. 7 5 1
9. 10 4
6
1) 2)
3)
10. 10
7 7 7
2
11. 7 1
4
4
1) 2)
12. 5 5
5 3 2
5
1)
2)
3)
13. 3 “ COLLEGE “
14. x y . . . 1 0 < y
x < 1 xy = 100!
15. m n n
m =
2
0
33 +
2
1
33+
2
2
33 + … +
2
33
33
m + n
“ COLLEGE ”
⌦⌦ 93
8
( binomial theorem ) 6
4
*****************************************************************************************
1.
1.1
1.2 (a + b)n
1.3 r (a + b)n
1.4 (a + b)n
1.5
2.
(a + b)n a , b
n
a , b n , r 0 r n
( a + b )n = an + n
1 an-1b +
n
2 an-2b2 + ..... +
n
r an-rbr + ..... +
n
n 1 abn-1 + bn
3.
4.
1 - 2
4.1 (a + b)2 (a + b)3 .3
4.2 (a + b)4 (a + b)5
( ) (a + b)n
n (a + b)10
(a + b)n a , b n
⌫ ⌫ ⌦⌫ ⌫⌫94
4.3 (a + b)n (a + b)n = (a + b)(a + b)
(a + b) ....... (a + b) n -
a
an b 1
a b
1 a n - 1 an-1b n
1
b an-1b n
1 b 2
a b
2 a n - 2 an-2b2 n
2
b an-2b2 n
2 b
bn
(a + b)n
n
r
4.4 (a + b)2 , (a + b)3 , (a + b)4 , (a + b)5
4.5 (a + b)n
(a + b)n a , b
(a + b)n
3 - 4
4.6 8
4.7
(a + b)n ( ) n
r an-rbr
(a + b)n ( r + 1)
(a+ b)n
(a + b)n
8
(1.2) 6 2
⌦⌦ 95
1.2 6
1.2 a + b
(1.2) 6
8
5. /
5.1 8
5.2 8
5.3 8
6.
7.
7.1
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.2
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
7.3
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
8.
…………………………………………………………………………………………………………...………...
……………………………………………………………………………………………………………………..
.…...………………………………………………………………………………………………………………..
…………………………………………………………………………………………………………...………...
⌫ ⌫ ⌦⌫ ⌫⌫96
8
******************************************************************************************************
(binomial theorem )
a , b ( a + b )2 = a2 + 2ab + b2
( a + b )3 = a3 + 3a2b + 3ab2 + b3
( a + b )2 = (a + b)(a + b)
a .............. a2
a
b .............. ab
a .............. ab a2 + 2ab + b2
b
b .............. b2
( a + b )3 = (a + b)(a + b)(a + b)
a ..................... a3
a
b ..................... a2b
a
a ..................... a2b
b
b ..................... ab2
a ..................... a2b a3 + 3a2b + 3ab2 + b3
a
b ..................... ab2
b
a ..................... ab2
b
b ..................... b3
⌦⌦ 97
( a + b )n n (a + b)
n n
( a + b )n n
( a + b )n = (a + b)(a + b)(a + b) ........... (a + b) n
a an an
b 1 a n - 1 an-1b
n
1 = n b an-1b
n
1
b 2 a n - 2 an-2b2
n
2 b 2 an-2b2
n
2
:
:
b bn bn
( a + b )n ( a + b )n an-rbr r
0 r n b r a n - r n
r
b r n
( a + b )n
n
r 0 r n ( binomial coefficient )
a , b n , r 0 r n
( a + b )n = an + n
1 an-1b +
n
2 an-2b2 + ..... +
n
r an-rbr + ..... + n
n 1 abn-1 + bn
⌫ ⌫ ⌦⌫ ⌫⌫98
( a + b )2 = a2 + 2
1 a2-1b + b2 = a2 + 2ab + b2
( a + b )3 = a3 + 3
1 a3-1b +
3
2 a3-2b2 + b3 = a3 + 3a2b + 3ab2 + b3
( a + b )4 = a4 + 4
1 a4-1b +
4
2 a4-2b2 +
4
3 a4-3b3 + b4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
1. ( a + b )n n + 1
2. ( a + b )n an
( a + b )n bn
n
0
n
n
n
0 = n
n = 1
3. ( a + b )n a n 1 0 b
0 1 n
4. ( a + b )n a b n
5. ( a + b )n
n
0 + n
1 + n
2 + ..... +
n
n = 2n
1 ( 2a + b )4
( 2a + b )4 = (2a)4 + 4
1 (2a)3 b +
4
2 (2a)2 b2 +
4
3 (2a) b3 + b4
= 16a4 + 4 (8a3) b + 6 (4a2) b2 + 4 (2a) b3 + b4
= 16a4 + 32 a3b + 24 a2 b2 + 8 a b3 + b4 Ans.
2 ( x - 1x
)6
(x - 1x
)6 = x6+ 6
1
1 6
2
1 6
3
1 6
4
1 6
5
1 15 4 2 3 3 2 4 5 6x
xx
xx
xx
xx
x x( ) ( ) ( ) ( ) ( ) ( )
= x6 - 6 x4 + 15 x2 - 20 + 15 x-2 - 6 x-4 + x-6 Ans.
⌦⌦ 99
( a + b )n
( a + b )n = an + n
1 an-1b +
n
2 an-2b2 + ..... +
n
r an-rbr + ..... +
n
n 1 abn-1 + bn
( a + b )n n
r an-rbr r + 1
( a + b )n
r + 1 ( a + b )n
3 5 ( 2x - 3x
)7
5 ( 2x - 3x
)7 = 7
42
33 4( ) ( )x
x
= 35 (8x3)81
4x
= 22,680 x-1 Ans.
4 ( x2 - y )11 x10
x10 r + 1
r + 1 ( x2 - y )11 = 11 2 11
rx y
r r( )
= 11 22 2
rx y
r r( )
x10 22 - 2r = 10 r = 6
x10 = 11
6
10 6x y( )
= 462 x10 y6 Ans.
r + 1 = n
r an-rbr
⌫ ⌫ ⌦⌫ ⌫⌫100
8
*****************************************************************************************
1. (1) (a + b )6 (2) (x - 2y) 5
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
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……………………………………………………………………………………………………………………
2. 4 ( a + 2x )10
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
3. ( x2 - y )11 y7
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
4. ( x2 + 1x
)12
(1) x9 (2) x-3
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
⌦⌦ 101
5. x ( x2 - 1x
)9
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
6.
(1) ( a + b )4 + ( a - b )4 (2) ( a + b )7 - ( a - b )7
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
7. (2 - x ) + (2 - x )2 + (2 - x )3 + (2 - x )4 + (2 - x )5
(1) x (2) x3/2
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
8. 2202 3 5
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………
*******************************************
⌫ ⌫ ⌦⌫ ⌫⌫102
8
******************************************************************************************************
1.
1) (2a + b)5
2) (2x - 3y)4
3) ( a2
+ 2b)6
4) ( 1a
- a2 )7
5) ( yy
1 )6
2.
1) 6 (x - y)10
2) 5 (x3 + 2)8
3) 10 (tan2A - 1)15
4) 8 ( x2
2 - 2y)16
5) ( xx
1 )14
6) a3 ( a2 + 1a
)12
7) y4 ( xy - 12x
)6
8) x ( xx
1
32 )10
9) y ( 5 32yy
)6
3. (2x - 3y)9
1) x6y3
2) x4y5
3)
4. ( a2 + 2a
)10
1) a2
2) a-1
3)
⌦⌦ 103
5. ( 3 - 2 x ) + ( 3 - 2 x )2 + ( 3 - 2 x )3 + ( 3 - 2 x )4
1) x
2) x3/2
6. ( x + x 1)5 + ( x - x 1)5
7. ( x 1 + x )5 - ( x 1 - x )5
8. 3
1) ( 1.01 )8
2) ( 1.1 )10
3) ( 0.98 )9
4) ( 1.99 )4
9. 12
1
99 + 2
2
2
99 + 3
2
3
99 +... + 99
2
99
99
10. 1 – x + x2 – x
3 + … + x
2006 – x
2007 = a0 + a1 y + a2 y
2 + … + a2007 y
2007 y = x + 1 a0, a1,
a2, …, a2007 a2
⌫ ⌫ ⌦⌫ ⌫⌫104
:
.
. .
.
.
:
:
.
:
:
:
:
:
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