Binomial Coefficient Definition of Binomial coefficient For nonnegative

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. 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-1 y , x n-2 y 2 , x n-3 y 3, … 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 = n. Co an bo +n. C 1 an-1 b 1 +n. C 2 an-2 b 2+…. . +n. Cn a 0 bn.

Example Write the binomial expansion of (x+y) 7 . Solution : Use the binomial theorem A=x; b=y; n=7 (x+7)7=x 7+7 c 1 x 6 y 1+7 c 2 x 5 y 2+7 c 3 x 4 y 3+7 c 4 x 3 y 4+7 c 5 x 2 y 5+ 6+ c y 7 c xy 7 6 7 7 Answer =x 7+7 x 6 y 1+21 x 5 y 2+35 x 4 y 3+35 x 3 y 4+21 x 2 y 5+7 xy 6+y 7

Question 2 (2 x-y) 4 Solution : Use the binomial theorem a=2 x; b=-y; n=y = (2 x) 4=4 c 1 (2 x) 3 y+4 c 2 (2 x) 2 y 2 -4 c 3 (2 x) y 3+4 c 4 y 4 Answer =16 x 4 -32 x 3 y+24 x 2 y 2 -8 xy 3+y 4

Question 3 (11)5= (10+1)5 Solution : Use the binomial theorem, to find the value of A=10; b=1; n=5 =105+5 c 1104 (1) +5 c 4103 (1)2+5 c 3 (10)2(1)3+5 c 4 (10)5 -4(1)4+5 c 5 (1) =100000+5 x 100000+10 x 1000+5 x 10+1 x 1 Answer =161051.

GENERAL TERM IN A BINOMIAL EXPANSION • • For n positive numbers we have (a+b)n = n. Co an bo +n. C 1 an-1 b 1 +n. C 2 an-2 b 2+…. . +n. Cn a 0 bn. According to this formula we have The first term=T 1= n. Co an b 0 The second term =T 2= n. C 1 an-1 b 1 The third term=T 3= n. C 2 an-2 b 2 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)