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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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