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0-2668-7974 0-2668-7123 2530

0-2243-1129, 0-2668-7329

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371.95

: 2552

ISBN 978-974-559-685-6

2. - 3.

580 8 . 34 1

. 10230

0-2943-8373-4 0-2510-7753

. .2542 ( 2) . .2545

( 2547)

(Acceleration Program)

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)

School in school Program 2547

Gifted Education

(Acceleration Program)

(Enrichment Program)

( 3 6 5

(Extension Program) (mentor)

AP Program (Advanced Placement Program)

(Acceleration

Program) 5 ( 6 )

Pre-test Post-test

11. 2 3

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

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1.2 1 2

1.3 1 2

3.3 1 2

4.3 1.1

4.4 1.1 1

4.5 1.1

4.7 1.2

4.8 1.2 2

4.9 1.2

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4.10 1

4.12 1

5.1 1.1 1.2 (1) (2)

5.2 1.1 1.2 (1) (2)

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1.1 (1)

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6 x 3 = 18 Ans.

2 1 2 2

2 x 2 = 4

1 2 1 n1

1 2 n2

H (H,H)

T (H,T)

H (T,H)

T (T,T)

4 (H,H) , (H,T) , (T,H) (T,T) Ans.

⌫ ⌫ ⌦⌫ ⌫⌫4

(1) 1 4 4

4 x 4 = 16 Ans.

(2) 1 4 4

4 x 3 = 12 Ans.

4 2 10

2 2 1 2

(1) 1 1 10

2 2 10

10 x 10 = 100 Ans.

(2) 1 1 10

10 x 9 = 90 Ans.

= 100 - 90

= 10 Ans.

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1.1 (1)

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1. ……………

1 …………………………………………………………………………………………………...

2 ……………………………………………………………………………………………………

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2. …………………….

1 …………………………………………………………………………………………………………

2 …………………………………………………………………………………………………………

3. ( 1)

………………..

1 …………………………………………………………………………………………………………

2 ………………………………………………………………….……………………………………..

4. 2 3

…………………

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1 1 ( ) 2 ( ) …….

2 1 2 …… x .…..

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2 Lee 3

A Z , Ten Co

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2. 1 1

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3. 2 , 4 , 6 , 7 9

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5. 14 26 1

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1 n1 1

2 n2 ………...…..

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1.2 (2)

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1 6 1, 2 , 3 , 4 , 5 6

2 1 2 6

3 2 3 6

6 x 6 x 6 = 216 Ans.

(1) 1 1 5

5 x 5 x 5 x 5 = 625 Ans.

(2) 1 1 5

1 n1 1

2 n2 2 3 n3

k n1 n2 n3 . . . nk

⌦⌦ 9

5 x 4 x 3 x 2 = 120 Ans.

7 6 1 6

6 7 x 6 x 5 x 4 x 3 x 2 = 5,040 Ans.

4 0 , 1 , 2 , 3 3

(1) 1 0 3

3 x 4 x 4 = 48 Ans.

(2) 1 0 3

3 x 3 x 2 = 18 Ans.

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1.2 (2)

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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 …………

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……….. 3

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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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k 1 n1

1 2 n2

2 3 n3 k nk

k ………………………………………….

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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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8. 4 1 5 2

8 3 3 4 6

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9. 100 999

(1) (3)

(2) (4) (ent.)

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1. 2 9

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

15. 1 2,000 1

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( factorial ) 6

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1.1 n n

n n x (n -1) x (n - 2) x ..... x 3 x 2 x 1

4.1 10

10 10 x 9 x 8 x . . . x 3 x 2

10 x 9 x 8 x . . . x 3 x 2 x1จากนั้นครูชี้แจงใหนักเรียนทราบวา จำนวนที่อยูในรูปลักษณะดังกลาวนี้ สามารถเขียนแทนดวยสัญลักษณ

⌫ ⌫ ⌦⌫ ⌫⌫16

4.3 n n

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….……………………..…………………………………………………………………………………………

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⌦⌦ 17

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( 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) !

1 ! = 1 (1 - 1) !

1 = 1 ( 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

1 2 3 4 5 6 7 8 = 8 x 7 x 6 x 5 x 4 = 6,720

9! 10 11 12 = 12 x 11 x 10 = 1,320

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

2 = 720

n + 1)(n)(n - 1) = 720

(n + 1)(n)(n - 1)

10 9 8 = 720

(n + 1) ! = 10! Ans.

( )( )( ) !

n n( )( ) = 1

3n = (n - 4)( n - 5)

n2 - 12 n + 20 = 0

( n - 10)(n - 2) = 0

n = 10 , 2

n = 10 Ans.

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

100 240

5 = 1! + 2! + 3! + 4! + 5! = 1 + 2 + 6 + 24 + 120 = 153

100 240 153 Ans.

⌫ ⌫ ⌦⌫ ⌫⌫20

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(1) 5 ! + 7 ! (2) 10! - 8! (3) 11

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(1) ( ) ! ( )

(2) ( ) ( )

( ) !( ) !

1 3 (3)

)!1( )1( )32(

)!( )!2(2

………………………………………………………………………………………………………………

(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 ( )

= 15,120

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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 = ( ) !

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⌫ ⌫ ⌦⌫ ⌫⌫22

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(1) 84

(2) 711

(3) 2 86

(4) 107

(5) nn ( )5

(6) ( )

(7) ( )

1 (n +1)

)2 (8)

n (n +1)

1 (9) n

n k ( )

(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 )

(1) ( )

= 1680 (2) 2 .

(3) ( )n

(4) 3 .

n = 2 .

(5) nn

= 30 . nn

5 ( )5

⌦⌦ 23

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2.2 n n

2.3 n r

P(n , r) = n

3.1 ( 3)

3.2 ( 4)

⌫ ⌫ ⌦⌫ ⌫⌫24

4.2 3.1

4.4 3.1

4.7 3.2

4.9 3.2

4.11 3

5.1 3.1

5.2 3.1

5.4 3.2

5.5 3.2

⌦⌦ 25

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( permutation )

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 !

7! = 5,040 Ans.

⌦⌦ 27

2 “SECOND”

2 x 1 x 4! = 48 Ans.

(1) 3 (2) 3

(1) 3 3

1 7 7!

2 3 3!

7! x 3! = 5,040 x 6 = 30,240 Ans.

(2) 3 (1)

3 = 9! - 7! 3! = 332,640 Ans.

⌫ ⌫ ⌦⌫ ⌫⌫28

(3) (2) 3 2

1 6 6!

2 3 7 7 x 6 x 5 = 210

= 6! x 210 = 151,200 Ans.

(1) 1 (2) 2 (3) 3

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!)

6 6 = 6!

= 6! 6!

2 6! 6! + 6! 6! = 2 (6! 6!) Ans.

⌦⌦ 29

(2) (1) 2 2

6 6 6!

= 6! 6!

2 6! 6! + 6! 6! = 2 (6! 6!) Ans.

(3) (1) (2) 3 2

6 6 6!

= 6! 6!

2 6! 6! + 6! 6! = 2 (6! 6!) Ans.

2 x n ! x n !

5 6 2 3

6! = 720 Ans.

⌫ ⌫ ⌦⌫ ⌫⌫30

6 6 12

(1) (2)

2 (6! 6!) Ans.

⌦⌦ 31

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1. 3 3 ………………………………………

2. 1 ……………………………………………..…………

3. 10 …………………………………………..

4. 3 ……………………………………………..…………

5. 9 ……………………..

6. 5 …………………… 1, 2, 3, 4, 5

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7. 7 …………………………………………………..……

8. 7 ……………………………

( ) ………………………………… 7

10. 6 6

……………………………………………………

11. 10 ( ) ……………………………………….

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12. 10 1 ………..…......

( ) ……………………. 10

( ) …………………………………………….………………

14. 13 ( ) ………………

” ……………………

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17. 16 n n

……………………………………… ( 3 )

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1. “NUMBER”

(1) (2)

(3) (4)

……………………………………………………………………………………………………………………

(1) (2)

(3) (4)

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⌦⌦ 33

3. 5 4 9

(1) 2 (2) 2

……………………………………………………………………………………………………………………

4. 1 , 2 , 3 , ..... , 9 1

1 , 2 3

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5. 4 1 , 2 , 3 4

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⌫ ⌫ ⌦⌫ ⌫⌫34

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1. 8 5 5

8 x 7 x 6 x 5 x 4

= 8 (8 -5)

2. 4 2 , 4 , 5 , 7 , 8 , 9

6 x 5 x 4 x 3

P(n , r)

1) r = n

P(n , n) = n (n - n)

2) nPr nPr Pn,r P(n , r)

n(n - r)

( r n )

P(n , r) = n(n - r)

⌦⌦ 35

1 P(4 , 2) = 4

! = 4 x 3 = 12

P(9 , 3) = 9

! = 9 x 8 x 7 = 504

P(15 , 5) = 15

= 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)

( )( )( ) !

( )( )

n2 - 12n + 20 = 0

( n - 10 )( n - 2 ) = 0

n = 10 , 2

n = 10 Ans.

3 10 3

P(10 , 3) = 10

= 10(9)(8)

= 720 Ans.

4 5 3 1 2

P(5 , 3) = 5

= 5(4)(3)

= 60 Ans.

⌫ ⌫ ⌦⌫ ⌫⌫36

5 5 “PERMUTATION”

(1) (2)

(3) (4) 1

“PERMUTATION” 10 5

(1) P(10 , 5) = 10

= 10(9)(8)(7)(6)

= 30,240 Ans.

4 P(9 , 4)

5 x P(9 , 4) = 5 x 9

= 5(9)(8)(7)(6) = 15,120 Ans.

4 P(9 , 4)

5 x P(9 , 4) = 5 x 9

= 5(9)(8)(7)(6) = 15,120 Ans.

= 5! = 120

1 = 30,240 - 120 = 30,120 Ans.

⌦⌦ 37

5 1 2 3

1 1 5 5

2 2 P(5 , 2) = 20

3 3 P(5 , 3) = 60

4 4 P(5 , 4) = 120

5 5 5! = 120

5 + 20 + 60 + 120 + 120 = 325 Ans.

7 1 , 2 , 3 , 4 , 5 , 6 4

4 = P(6 , 4) = 6

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

10 , 100 1,000

1,260 + (1,260 x 10) + (1,260 x 100) + (1,260 x 1,000) = 1,399,860 Ans.

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⌫ ⌫ ⌦⌫ ⌫⌫38

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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

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8 …………………………………….

11. 10 ( ) …………………….…………………

………………………………………………………………………………......

12. 10 1 ……………..

( ) ……………………. 10

13. ( ) …………………………

14. n r

……………………………………………. ( 4 )

15. n r

P(n , r) P(n , r) ……………………………………………………

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

⌦⌦ 39

******************************************************************************************************

1. 7 A , B , C , D , E , F G

3. 4 2

4. 3 4 2

6. 12 12

(1) (2) 1

(3) 2 (4) 3

7. 6 6

(1) (2)

8. 4 3 3

(1) (2) 3

9. 8 4

(1) 4 (2) 3

(3) 2 (4) 3

10. 8 3

(1) (2)

(1) (2) 2

(1) 6 (2) 7

13. A = {1, 2, 3, 4, 5, 6 } B = { {1} , {1, 2}, {1, 2, 3 } } f : B A f (x) x

14. 4,000,000 250

15. A = { 1, 2 , 3, 4, 5, 6 } B = { 1, 2, 3 } f : A B

⌫ ⌫ ⌦⌫ ⌫⌫40

*****************************************************************************************

1.2 2 n

1.3 2 n

1.5 3 n

1.6 3 n

2.2 2 2

2.3 n 2 n

= ( n - 1)!

2.4 n 2 r

= P n r

( , ) !

3.2 2 ( 5 6)

3.3 3 ( 7 8)

⌦⌦ 41

4.10 4

4.11 4 5 6

4.12 4.1

4.13 4

4.14 4.2

4.15 4.2 ( )

⌫ ⌫ ⌦⌫ ⌫⌫42

4.16 7 8

5.1 4 2

5.2 4.1 2

5.3 4 2

5.4 4.2 3

…………………………………………………………………………………………………………………......

……………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………………......

……………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………………......

……………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………………......

……………………………………………………………………………………………………………………..

⌦⌦ 43

****************************************************************************************************

1. n = 2 …………………………………

………………………………………………………………..……

2. n = 3 …………………………………

……………………………………………………………………..

3. n = 4 …………………………………

……………………………………………………………………..

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 …….…

5.1 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

******************************************************************************************************

A , B , C 3 ! = 6

ABC BCA CAB

ACB CBA BAC

ABC BCA CAB

ACB CBA BAC

A , B , C

3 3! 33 ! = 2!

⌫ ⌫ ⌦⌫ ⌫⌫46

4 A , B , C , D 4 ! = 24

ABCD BCDA CDAB DABC

4 4 ! 44 ! = 3 !

n - 1 (n - 1 ) !

n r P(n , r)

P(n , r) P n r

( , ) = n

n r! ! r( )

2 (n - 1 ) !

2 r P n r

⌦⌦ 47

1 6 4 2

(1) (2) 2

(3) 2 (4) 2

6 (6 - 1)! = 5! = 120 Ans.

(2) 2 2 1

5 (5 - 1)! = 4!

4! x 2! = 24 x 2 = 48 Ans.

120 - 48 = 72 Ans.

n n - 1

1 x 4! = 24 Ans.

(1) (2) , ,

(3) , , (4)

(1) 8 8 2

(8 - 1)! = 7! = 5,040 Ans.

(2) , 1

5! x 3! = 120 x 6 = 720 Ans.

⌫ ⌫ ⌦⌫ ⌫⌫48

(3) , , 3

, , 5 P(5 ,3)

4! x P(5 ,3) = 24 x 5 x 4 x 3 = 1,440 Ans.

2 x 5 x 5! = 1,200 Ans.

3 10 5

p( , ) !

= 2(9)(8)(7)(6) = 6,048 Ans.

#####################################

⌦⌦ 49

******************************************************************************************************

1. 4 2 2

……………………………………………………………………………………………………………………

2. 4 2

……………………………………………………………………………………………………………………

3. 6 6

1) 1 2) 2 3) 3

……………………………………………………………………………………………………………………

⌫ ⌫ ⌦⌫ ⌫⌫50

4. 1 3 3

……………………………………………………………………………………………………………………

6. 15 10

……………………………………………………………………………………………………………………

***********************************************

⌦⌦ 51

******************************************************************************************************

2 A , B , C , D 3 ! = 6

( )n 1

⌫ ⌫ ⌦⌫ ⌫⌫52

= ( ) ! !7 1

2 = 360 Ans.

2 9 2 3

(1) (2)

(1) 9 9

= ( ) ! ! ,9 1

40 320

= 20,160 Ans.

6 = ( ) ! !6 1

2 = 60

2 2! = 2

3 3! = 6

= 60 x 2 x 6 = 720 Ans.

3 10 6

= P( , )

( )( !)

2 6 4 = 12,600 Ans.

@@@@@@@@@@@@@@@@@@@@@@@@@

8 n 3 r

⌦⌦ 53

******************************************************************************************************

2. 4 3

(1) (2)

(1) (2) 1

4. 5 1

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)

(1) (2) (3)

10. 10

⌫ ⌫ ⌦⌫ ⌫⌫54

*****************************************************************************************

2.1 n k 1 n1 2

n2 . . . k nk n1 + n2 + ..... +nk = n

n n !n1! n

k ! ..... !

2.2 n k 1 n1

2 n2 ........ k nk n1 + n2 + ..... + nk = n

n !n1! n

k ! ..... !

2.3 n k 1 n1 2 n2

............. k nk n1 + n2 + ..... + nk = n . . (n1 , n2 , ..... , nk )

= 1 n ( )n 1 !

k ! ..... !

⌦⌦ 55

3.1 ( 9)

3.3 ( 10)

4.1 A , B , C

4.2 A B M

4.3 3 M , M , C

4.4 “ELEMENT” E

4.9 5.2

⌫ ⌫ ⌦⌫ ⌫⌫56

5.1 5.1

5.3 5.2

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

⌦⌦ 57

******************************************************************************************************

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

1 (MMC) 2 !

3 3 x 2 ! = 3 ! = 6

3 3 !2 !

2. ”ELEMENT”

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 4 2 3

! ! ! !

= 277,200 Ans.

2 4 4 3

(1) (2)

(1) 11 4

= 1,663,200 Ans.

n1 + n2 + ..... +nk = n

n n !n1! n

k ! ..... !

⌦⌦ 59

(2) 11 3

= 3! 4! 3! = 864 Ans.

! = 1,680

8 3 9 P(9 , 3) = 9

! = 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

! ! = 420 Ans.

(2) 2 3 4 2

2 2 1 2

! ! = 180

7 1 x 180 = 180 ***

2 3 4 2

2 3 1 2

! ! = 60

7 2 x 60 = 120 ***

7 2 180 + 120 = 300

9 “ ” ( partitioning law )

n k 1 n1

2 n2 ........ k nk n1 + n2 + ..... + nk = n

n !n1! n

k ! ..... !

⌫ ⌫ ⌦⌫ ⌫⌫60

( !) !

4 10 2

(2) 1 10

(1) 10 3 2 , 3 5

! ! ! = 2,520 Ans.

(2) 9 3 2 ,3 4

! ! ! = 1,260 Ans.

5 10 2 2

! ! = 252 Ans.

* 6* 10 2

! ! ! = 126 Ans.

* 7* 9 3 4 3

1 9 3 2 , 3 4 9

! ! ! = 1,260

2 3 3 3! = 6

1,260 x 6 = 7,560 Ans.

@@@@@@@@@@@@@@@@@@@@@@@

⌦⌦ 61

******************************************************************************************************

1. “MISSISSIPPI”

……………………………………………………………………………………………………………………

2. 3 2 4

(1) (2) 2

……………………………………………………………………………………………………………………

3. 9 2,520

……………………………………………………………………………………………………………………

⌫ ⌫ ⌦⌫ ⌫⌫62

(3) C D

(4) C D

……………………………………………………………………………………………………………………

5. 9 3 4 3 2

……………………………………………………………………………………………………………………

⌦⌦ 63

6. 10 3

(1) 4 2 3

(2) 4 2 3

……………………………………………………………………………………………………………………

3 4 3 3

(1) (2)

……………………………………………………………………………………………………………………

8. 6 3 3

……………………………………………………………………………………………………………………

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

⌫ ⌫ ⌦⌫ ⌫⌫64

******************************************************************************************************

: 1. 10 n1 , n2 , ..... , nk 1 . .

(n1 , n2 , ..... , nk ) = 1

2. . . (n1 , n2 , ..... , nk ) 1 (

) AABBCC

( . . . 2 , 2 , 2 2) 16

2 2 2 = 15

(Mobius Theorem)

1 2 3 4

4 . . . 2 , 3 4 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 !

k ! ..... !

⌦⌦ 65

2 1 2 2

(1) (2) (3) 3

(1) 8 2 2

. . . 2 , 2 , 3 1 1

2 2 3 = 210 Ans.

3! = 6 Ans.

(3) 1 6 6

2 2 1 1

. . . 2 , 2 , 1 1 1

2 2 = 30 Ans.

B 2 W 2

BB WWGGGG

. . . 1 , 2 4 1

2 4 = 15

15 Ans.

BB WWGGGG

⌫ ⌫ ⌦⌫ ⌫⌫66

4 3 8 6

B 6 W 2

. . . 6 2 2

4 Ans.

. . . 1 , 2 2 1

= ( ) !

2 2 = 6

= 3 x 6 = 18 Ans.

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

แบบที ่1W ตดิกนั 2 ชอง

แบบที ่2W 1 ชอง เวน 1 ชอง

⌦⌦ 67

******************************************************************************************************

1. “COOPERATOR”

(1) (2) O 3

(3) R 2

2. “MATHEMATICIAN”

(1) A 3 (2) A 2

3. “TROTTING”

4. 4 3

(1) (2) (3)

5. 7 2 2 3

6. 2 3 2

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

⌫ ⌫ ⌦⌫ ⌫⌫68

13. 11

(1) 4 4

(2) 2 3 3

(3) 6 1 1

15. 9 3

(1) (2) 3

(3) 1 (4) 1

16. 10

12,600

17. 5 3

18. “PARALLEL”

19. X, X, Y, Y, Y, Z, Z

(1) (2) X 2

(3) Y 3 (4) Z

20. 2 4 3

⌦⌦ 69

*****************************************************************************************

1.3 r n

1.4 r n

2.1 r n

2.2 n r n

! ( ) !

3.1 r n ( 11)

4.1 A , B , C , D 3 4

ABC , ABD , ACD , BCD

4 A , B , C , D

4.3 3 4

⌫ ⌫ ⌦⌫ ⌫⌫70

1) 3 4

( 3 ! )

2) 4 ( 3 ! x 4 )

4 [ P(4 , 3) ]

4 x 3! = P(4 , 3)

4 = !3

)3,4(P

3 4 = !3

)3,4(P

r n ( 11)

4.7 C (n , r) C (n , r) n r

4.9 6.1

4.10 6.1

4.11 r

4.12 6

4.14 1 ( )

4.16 r

4.17 6.2

⌦⌦ 71

4.18 r

5.1 6.1 r n

5.2 6 r n

5.3 6.2 r n

5.4 ( )

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

⌫ ⌫ ⌦⌫ ⌫⌫72

6.1 r n

******************************************************************************************************

( combination )

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

11 n r

nr n r

! !! ( )

r = nr n r

! !! ( )

⌦⌦ 73

!( ) !

10 . . .

8 . . .

20 1 2 3

. . . .

n n n n n

⌫ ⌫ ⌦⌫ ⌫⌫74

(1) 1 2

! ! = 45 Ans.

(2) 1 3

! ! = 120 Ans.

(1) 2 12

(2) 12 A , B , C D

4 3 2 12

(1) 12 3

! ! = 66 Ans.

(2) 12 4

! ! = 6

2 - 6 + 1 = 61 Ans.

⌦⌦ 75

4 2 6 7 2

! ! = 15

! ! = 21

15 x 21 = 315 Ans.

(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

! ! = 2,598,960 Ans.

(2) 1 4 4 4

1 = 48

1 x 48 = 48 Ans.

(3) 1 13 13

! ! = 78

! ! = 286

2 3 78 x 286 = 22,308 Ans.

⌫ ⌫ ⌦⌫ ⌫⌫76

5 13 13

! ! = 1,287

5 4 x 1,287 = 5,148 Ans.

(1) 3 9

3 = 84

3 = 20

3 84 x 20 = 1,680 Ans.

(2) 1 , 2 , 3 , 4 , 5 6

6 = 5,004

5,004 Ans.

7 5 2 1 1

(1) 2 2 (2)

(4) 2 2 1

(1) 2 5 5

2 = 10

2 2 10 x 10 = 100 Ans.

⌦⌦ 77

(2) 4 4 4

5 4 5 5

1 = 16

5 x 16 = 80 Ans.

(3) 1 1 5

2 2 2 2 4

1 5 x 6 x 4 = 120 Ans.

5 x 4 x 3 = 60 ***

2 = 10 ***

2 60 + 10 = 70 Ans.

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

⌫ ⌫ ⌦⌫ ⌫⌫78

******************************************************************************************************

1. C (n , r) = C (n , n - r)

……………………………………………………………………………………………………………………

2. r C (n , r) = P (n , r)

……………………………………………………………………………………………………………………

3. a b b 17

……………………………………………………………………………………………………………………

⌦⌦ 79

5. .6 9

……………………………………………………………………………………………………………………

6. 1 1

……………………………………………………………………………………

…………………………………………………………………………………………………………………….

….…………………………………………………………………………………………………………………

…….………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………

……….……………………………………………………………………………………………………………

7. A = { a , b , c , d } ; B = { p , q , r , s , t } C B C A

……………………………………………………………………………………………………………………

*******************************************

⌫ ⌫ ⌦⌫ ⌫⌫80

6.2 r n

******************************************************************************************************

1 6 1 1 1 3

1 x 1 x 1 x 1 = 1

2 2 2 2

1 x 3 = 3

3 3 3 1

1 x 3 = 3

1 + 3 + 3 = 7 Ans.

⌦⌦ 81

2 a , a , a , b , b , c , c , d , d , e , f 4

4 = 15

2 = 10

4 x 10 = 40

1 x 5 = 5

15 + 40 + 6 + 5 = 66 Ans.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

⌫ ⌫ ⌦⌫ ⌫⌫82

******************************************************************************************************

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

2) 2 3 1 2

5. 5 10 2

1) 3 1 1

2) 3 3) 3

6. 3 20

2) 20 2

8. 10 6

1) 2 2) 3

9. 10 A , B C

1) 2) A

3) A B 4) A B C

10. 10 4

⌦⌦ 83

13. 20

15. 6 4

1) 4 2) 4 1

3) 4 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

18. 2 2 3 3 3

1) 4 2) 2 3) 2

19. 8 6 2

1) 4 2) 4

⌫ ⌫ ⌦⌫ ⌫⌫84

*****************************************************************************************

1 “ 6 6

( 20 )

( 40 )

⌦⌦ 85

3 – 5

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

⌫ ⌫ ⌦⌫ ⌫⌫86

******************************************************************************************************

1 4 8 8

2 3 9 9

3 7 7 7!

3 x 7! Ans.

1 2 5 5

2 2 5 5

3 2 2 4!

2 x 4! = 10 x 10 x 24

= 2,400 Ans.

3 10 6 4 3

1 3 6 6

2 2 4 4

⌦⌦ 87

3 3 2 2

5! - 4! 2! = 120 - 48 = 72

2 x 72

= 20 x 6 x 72

= 8,640 Ans.

4 10 5 5

1 3 5 5

2 3 5 5

3 2! 3! = 12

3 x 12

= 10 x 10 x 12

= 1,200 Ans.

1 1 7 5

1 3 5 5

2 3 7 7

3 3 3 1 1

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 8 ( I , N , T , E , R , M , D , A ) 8

2 5 5!

5 x 5!

= 56 x 120 = 6,720

1 2 1 3 ( I , T , E) 3

2 3 7 ( 5 1

2 7 ) 7

3 5 2 5

= 3 x 35 x 60 = 6,300

3 5 2 2

1 2 2 3 ( I , T , E) 3

2 1 6 ( 5 2

1 6 ) 6

3 5 2 2 5

= 3 x 6 x 30 = 540

⌦⌦ 89

4 5 2 1 3 1

1 2 1 2 ( I , T) 2

2 3 1 (E) 1

3 5 2 1 3 1

1 x 1 x

= 2 x 1 x 10 = 20

1 3 1 (E) 1

2 2 7 ( 5 2 7

3 5 3 5

= 1 x 7

= 1 x 21 x 20

5 = 6,720 + 6,300 + 540 + 20 + 420

= 14,000 Ans.

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

⌫ ⌫ ⌦⌫ ⌫⌫90

******************************************************************************************************

1. 8 4 5

(1) 3 2 (2) 1

……………………………………………………………………………………………………………………

2. 5 6

……………………………………………………………………………………………………………………

⌦⌦ 91

4. 312 259

……………………………………………………………………………………………………………………

5. 5 5

……………………………………………………………………………………………………………………

6. 6 6 1 4 4

……………………………………………………………………………………………………………………

*****************************************

⌫ ⌫ ⌦⌫ ⌫⌫92

******************************************************************************************************

1. 10 3 3

2. 3 2 7

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

10. 10

11. 7 1

12. 5 5

13. 3 “ COLLEGE “

14. x y . . . 1 0 < y

x < 1 xy = 100!

15. m n n

33 + … +

“ COLLEGE ”

⌦⌦ 93

( binomial theorem ) 6

*****************************************************************************************

1.2 (a + b)n

1.3 r (a + b)n

1.4 (a + b)n

(a + b)n a , b

a , b n , r 0 r n

( a + b )n = an + n

1 an-1b +

2 an-2b2 + ..... +

r an-rbr + ..... +

n 1 abn-1 + bn

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 -

an b 1

1 a n - 1 an-1b n

b an-1b n

2 a n - 2 an-2b2 n

b an-2b2 n

(a + b)n

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

(a + b)n ( ) n

r an-rbr

(a + b)n ( r + 1)

(a+ b)n

(a + b)n

(1.2) 6 2

⌦⌦ 95

1.2 a + b

(1.2) 6

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

……………………………………………………………………………………………………………………..

.…...………………………………………………………………………………………………………………..

…………………………………………………………………………………………………………...………...

⌫ ⌫ ⌦⌫ ⌫⌫96

******************************************************************************************************

(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

b .............. ab

a .............. ab a2 + 2ab + b2

b .............. b2

( a + b )3 = (a + b)(a + b)(a + b)

a ..................... a3

b ..................... a2b

a ..................... a2b

b ..................... ab2

a ..................... a2b a3 + 3a2b + 3ab2 + b3

b ..................... ab2

a ..................... ab2

b ..................... b3

⌦⌦ 97

( a + b )n n (a + b)

( 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

1 = n b an-1b

b 2 a n - 2 an-2b2

2 b 2 an-2b2

b bn bn

( a + b )n ( a + b )n an-rbr r

0 r n b r a n - r n

( a + b )n

r 0 r n ( binomial coefficient )

a , b n , r 0 r n

( a + b )n = an + n

1 an-1b +

2 an-2b2 + ..... +

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 +

2 a3-2b2 + b3 = a3 + 3a2b + 3ab2 + b3

( a + b )4 = a4 + 4

1 a4-1b +

2 a4-2b2 +

3 a4-3b3 + b4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

1. ( a + b )n n + 1

2. ( a + b )n an

( a + b )n bn

3. ( a + b )n a n 1 0 b

4. ( a + b )n a b n

5. ( a + b )n

2 + ..... +

n = 2n

1 ( 2a + b )4

( 2a + b )4 = (2a)4 + 4

1 (2a)3 b +

2 (2a)2 b2 +

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

(x - 1x

)6 = x6+ 6

1 15 4 2 3 3 2 4 5 6x

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 +

2 an-2b2 + ..... +

r an-rbr + ..... +

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

5 ( 2x - 3x

)7 = 7

33 4( ) ( )x

= 35 (8x3)81

= 22,680 x-1 Ans.

4 ( x2 - y )11 x10

x10 r + 1

r + 1 ( x2 - y )11 = 11 2 11

r r( )

= 11 22 2

r r( )

x10 22 - 2r = 10 r = 6

x10 = 11

10 6x y( )

= 462 x10 y6 Ans.

r + 1 = n

r an-rbr

⌫ ⌫ ⌦⌫ ⌫⌫100

*****************************************************************************************

1. (1) (a + b )6 (2) (x - 2y) 5

……………………………………………………………………………………………………………………

2. 4 ( a + 2x )10

……………………………………………………………………………………………………………………

3. ( x2 - y )11 y7

……………………………………………………………………………………………………………………

4. ( x2 + 1x

(1) x9 (2) x-3

……………………………………………………………………………………………………………………

⌦⌦ 101

5. x ( x2 - 1x

……………………………………………………………………………………………………………………

(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

******************************************************************************************************

1) (2a + b)5

2) (2x - 3y)4

3) ( a2

+ 2b)6

4) ( 1a

- a2 )7

5) ( yy

1) 6 (x - y)10

2) 5 (x3 + 2)8

3) 10 (tan2A - 1)15

4) 8 ( x2

2 - 2y)16

5) ( xx

6) a3 ( a2 + 1a

7) y4 ( xy - 12x

8) x ( xx

32 )10

9) y ( 5 32yy

3. (2x - 3y)9

1) x6y3

2) x4y5

4. ( a2 + 2a

2) a-1

⌦⌦ 103

5. ( 3 - 2 x ) + ( 3 - 2 x )2 + ( 3 - 2 x )3 + ( 3 - 2 x )4

2) x3/2

6. ( x + x 1)5 + ( x - x 1)5

7. ( x 1 + x )5 - ( x 1 - x )5

1) ( 1.01 )8

2) ( 1.1 )10

3) ( 0.98 )9

4) ( 1.99 )4

99 + 2

99 + 3

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

ONEC Backoffice for Administratorbackoffice.onec.go.th/uploads/Book/481-file.pdf · Â µ¦ ´...

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