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Binomial Coefficient

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Definition of Binomial coefficient

For nonnegative integers n and r with n > r the expansion (read “n above r”) is called a binomial coefficient and is defined by

!

!( )!n r

n nC

r r n r

n

r

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Evaluating binomial coefficient

• Example

6 6! 6! 6 5 4 3 2 115

2 2!(6 2)! 2!4! 2 1 4 3 2 1

8 8! 8! 8!1

0 0!(8 0)! 0!8! 1 8!

!

!( )!n r

n nC

r r n r

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Your Turn

!

!( )!n r

n nC

r r n r

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Answer

5 5!

2 2! 5 2 !

5 4 3 2 1

2 1 3 2 1

20

102

!

!( )!n r

n nC

r r n r

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Expanding binomial

• The theorem that specifies the expansion of any power (a+b)n of a binomial (a+b) as a certain sum of products

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We can easily see the pattern on the x's and the a's. But what about the coefficients? Make a guess and then

as we go we'll see how you did. 0

1x a

1x a x a

2 2 22x a x ax a

3 3 2 2 33 3x a x ax a x a

4 4 3 2 2 3 44 6 4x a x ax a x a x a

5x a 5 4 2 3 3 2 4 5__ __ __ __x ax a x a x a x a

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Pascal’s Triangle

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Pascal’s Triangle

• Each row of the triangle begins with a 1 and ends with a 1.

• Each number in the triangle that is not a 1 is the sum of the two numbers directly above it (one to the right and one to the left.)

• Numbering the rows of the triangle 0, 1, 2, … starting at the top, the numbers in row n are the coefficients of x n, x n-1y , x n-2y2 , x n-3y3, … y n in the expansion of (x + y)n.

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Binomial Theorem

• The a’s start out to the nth power and decrease by 1 in power each term. The b's start out to the 0 power and increase by 1 in power each term.

• The binomial coefficients are found by computing the combination symbol. Also the sum of the powers on a and b is n.

(a+b)n = nCo an bo +nC1 an-1 b1 +nC2 an-2 b2+…..+nCn a0 bn.

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ExampleWrite the binomial expansion of (x+y) 7

. Solution :Use the binomial theorem A=x; b=y; n=7

(x+7)7=x7+7c1x6y1+7c2x5y2+7c3x4y3+7c4x3y4+7c5x2y5+ 7c6xy6+7c7y7

Answer

=x7+7x6y1+21x5y2+35x4y3+35x3y4+21x2y5+7xy6+y7

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Question 2(2x-y) 4

Solution :Use the binomial theorem a=2x; b=-y; n=y

= (2x) 4=4c1 (2x) 3y+4c2 (2x) 2y2-4c3 (2x) y3+4c4y4

Answer

=16x4-32x3y+24x2y2-8xy3+y4

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Question 3(11)5= (10+1)5

Solution : Use the binomial theorem, to find the value of A=10; b=1; n=5

=105+5c1104 (1) +5c4103 (1)2+5c3 (10)2(1)3+5c4 (10)5-4(1)4+5c5 (1)

=100000+5x100000+10x1000+5x10+1x1

Answer

=161051.

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GENERAL TERM IN A BINOMIAL EXPANSION

• For n positive numbers we have• (a+b)n = nCo an bo +nC1 an-1 b1 +nC2 an-2 b2+…..

+nCn a0 bn.• According to this formula we have • The first term=T1= nCo an b0

• The second term =T2= nC1 an-1 b1

• The third term=T3= nC2 an-2 b2

• So, any individual terms, let’s say the ith term, in a binomial

• Expansion can be represented like this: Ti=n C(i-1) an-(i-1) b(i-1)

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EXAMPLE•  

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MIDDLE TERM

•  

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• Find the middle term in the expansion of (4x-y) 8

Ti= th term =5th term

T5=8C4(4x)8-4(-y)4

T5= 70(256x4) (y4)

T5=17920x4y4

EXAMPLE

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Example•  

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Group Members

• Ayesha Khalid

• Hira Shamim Syed

• Urooj Arshad Syed

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