binomial expansion and more

Jeff Bivin -- LZHS

Binomial Expansion And More

Last Updated: May 2, 2011

Jeffrey Bivin

Lake Zurich High School

[email protected]

Jeff Bivin -- LZHS

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

(x + y)6 = 1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6

Let’s look at (x + y)p

(x + y)7 = _x7 + _x6y + _x5y2 + _x4y3 + _x3y4 + _x2y5 + _xy6 + _y7

Jeff Bivin -- LZHS

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

(x + y)6 = 1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6


Jeff Bivin -- LZHS

(x + y)7 = 1 7 21 35 35 21 7 1

(x + y)0 = 1

(x + y)1 = 1 1

(x + y)2 = 1 2 1

(x + y)3 = 1 3 3 1

(x + y)4 = 1 4 6 4 1

(x + y)5 = 1 5 10 10 5 1

(x + y)6 = 1 6 15 20 15 6 1

(x + y)8 = 1 8 28 56 70 56 28 8 1


Jeff Bivin -- LZHS

(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + 1y7

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

(x + y)6 = 1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6


Jeff Bivin -- LZHS

12! 47900160019,958,400

2! 3! 2! 24

In how many ways can you arrange the letters in the word MATHEMATICAL ?

Jeff Bivin -- LZHS

52 yx

35144

5040

!3!4

!7

21240

5040

!5!2

!7

34 yx

In how many ways can you arrange the letters in the non-word xxxxyyy?

In how many ways can you arrange the letters in the non-word xxyyyyy?

(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7

Jeff Bivin -- LZHS

Let’s look closer(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7

071!0!7

!7yx 167

!1!6

!7yx 2521

!2!5

!7yx

3435!3!4

!7yx 4335

!4!3

!7yx 5221

!5!2

!7yx

617!6!1

!7yx 701

!7!0

!7yx

Jeff Bivin -- LZHS

An alternate look(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7

77!0!7

!7C 67!1!6

!7C 57!2!5

!7C

47!3!4

!7C 37!4!3

!7C 27!5!2

!7C

17!6!1

!7C 07!7!0

!7C

Jeff Bivin -- LZHS

(2x - y)4 = 16x4 - 32x3y + 24x2y2 - 8xy3 + y4

40444 161)()2(1

!0!4

!4xyxC

)(84)()2(4!1!3

!4 31334 yxyxC

)(46)()2(6!2!2

!4 222224 yxyxC

)(24)()2(4!3!1

!4 33114 yxyxC

)(1)()2(1!4!0

!4 44004 yyxC

Jeff Bivin -- LZHS

(3x + 2y)5 = 243x5 + 810x4y + 1080x3y2 + 720x2y3 + 240xy4 + 32y5

50555 2431)2()3(1

!0!5

!5xyxC

yxyxC 2815)2()3(5!1!4

!5 41445

222335 42710)2()3(10

!2!3

!5yxyxC

323225 8910)2()3(10

!3!2

!5yxyxC

44115 1635)2()3(5

!4!1

!5yxyxC

55005 3211)2()3(1

!5!0

!5yyxC

Jeff Bivin -- LZHS

Given: (x + y)15

In how many ways can you arrange the

letters in the non-word

xxxxxyyyyyyyyyy ?

105515 3003

!10!5

!15yxC

1053003 yx

What is the coefficient of the term ____ x5y10 ?

Jeff Bivin -- LZHS

Given: (4x - 3y)10



xxxxxxxyyy ?

37710 )3()4(120

!3!7

!10yxC

37 2716384120 yx

What is the coefficient of the term ____ x7y3 ?

37160,084,53 yx

Jeff Bivin -- LZHS

Expand: (x + y + z)2

(x + y + z) (x + y + z)

x2 + xy + xz + yx + y2 + yz + zx + zy + z2

x2 + 2xy + 2xz + y2 + 2yz + z2

Jeff Bivin -- LZHS

x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xz2 + y3 + 3y2z + 3yz2 + z3

(x + y + z)2 (x + y + z)

x3 + x2y + x2z + 2x2y + 2xy2 + 2xyz + 2x2z + 2xzy + 2xz2 + y2x + y3 + y2z

+2yzx + 2y2z + 2yz2 + z2x + z2y + z3

Expand: (x + y + z)3

(x2 + 2xy + 2xz + y2 + 2yz + z2)(x + y + z)

We did this in the last example

Jeff Bivin -- LZHS

zyx 6!1!1!1

!3

x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xz2 + y3 + 3y2z + 3yz2 + z3

Given: (x + y + z)3

In how many ways can you arrange the letters in the non-word xyz ?

What is the coefficient of the term ____ xyz ?

In how many ways can you arrange the letters in the non-word xxz ?

zx 23!1!0!2

!3

What is the coefficient of the term ____ x2z ?

Jeff Bivin -- LZHS

Given: (x + y + z)15



xxyyyyyyyzzzzzz ?

672180180!6!7!2

!15zyx

672180,180 zyx

What is the coefficient of the term ____ x2y7z6 ?

Jeff Bivin -- LZHS

Given: (2x + 3y - z)9



xxxyyyyzz ?

243 )()3()2(1260!2!4!3

!9zyx

243480,816 zyx

What is the coefficient of the term ____ x3y4z2 ?

243 8181260 zyx

Jeff Bivin -- LZHS

Given: (a + b + c + d)20



aaaaabbbbbbcccccccdd ?

2765720,510,793,2!2!7!6!5

!20dcba

2765720,510,793,2 dcba

What is the coefficient of the term ____ a5b6c7d2 ?

Jeff Bivin -- LZHS

Binomial ProbabilityCan be determined in a binomial experiment that meets the following criteria:

► There are n independent trials.

► Each trial has only two possible outcomes:■ Success■ Failure

► The probability of success (s) is the same for each trial and the probability for failure (f) is the same for each trail.

knkkn fsCtrialsninsuccessesk Pr

Jeff Bivin -- LZHS

Binomial Probability

A die is rolled 5 times. What is the probability of rolling exactly 3 ones?

2653

61

35C

3888125

7776250

3625

216110

2

653

61

!2!3!5

Jeff Bivin -- LZHS


A bent coin has a probability of heads of 4/7. If the coin is tossed 10 times, what is the probability of tossing exactly 6 heads?

4736

74

610C

247.0210 249,475,282960,672,69

240181

1176494096

Jeff Bivin -- LZHS

A bent coin has a probability of heads of 4/7. If the coin is tossed 10 times, what is the probability of tossing at least 8 heads?


07310

74

10101

739

74

9102

738

74

810 CCC

126.0249,475,282976,454,35

07310

741

739

742

738

74 11045

Jeff Bivin -- LZHS

A bent coin has a probability of heads of 4/7. If the coin is tossed 10 times, what is the probability of tossing at least 3 heads?

8

732

74

2109

731

74

11010

730

74

0101 CCC

HHHHHHHHHHH 109876543210


980.0249,475,282960,904,276

8732

749

731

7410

730

74 451011

10

8291100

7

344534103411

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