Binomial Coefficient

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

Evaluating binomial coefficient Example

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Answer

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

We can easily see the pattern on the x's and the a's 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.

Pascal’s Triangle

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.

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.

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

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

Answer 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.

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)

EXAMPLE  

MIDDLE TERM  

EXAMPLE 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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