Please help me with this math problem

Question

Find the 9th term of (x-2y)^(10)

Answer 1

There are several different ways to go about answering this question; of course, they should all give the same answer.

The most direct way is to use a formula: (There are several variations in expressing this formula.)

The k-th term in the expansion of (a+b)ⁿ is found by evaluating the expression

n!/((n-k)!k!) * (a)^n-k * (b)^k with k ranging from 0 to n. Also the ORDER of the exponents in the terms needs to be considered. Here the terms would be ordered with the HIGHEST power of "a" first, aⁿ. (If you want the terms in the order where "a" starts with the LOWEST power, a⁰=1, then reverse the k and n-k everywhere in the expression.)

The "!" symbol represents "factorial".

In our problem, n=10, a=x, b= 2y. If we wanted the first term, we would use k=0; k=1 for the second term, and so forth. For the ninth term, k=8.

So n! / (( n-k) ! k!) * (a) ^n-k * (b) ^k =

10! / ((10-8)! * 8!) * (x) ^(10-8) * (2y)⁸ =

10! / (2! * 8!) * x² * 2⁸ y⁸ =

45 * x² * 256 y⁸ = 11520 x² y⁸ <--- ANSWER

A quick way to get the 45 is to use your C(n,r) key on your calculator. (Might be _nC_r there).

BUT, If you have to do it by hand,

10! / (2! * 8!) = 10*9*8*7*6*5*4*3*2*1 / (2*1 * 8*7*6*5*4*3*2*1) and the 8 down to 1 cancel in both places, so leaving 10*9 / (2*1) and that is easy to see is 45.

The 45 is the binomial coefficient for this term. But, 11520 is the coefficient of the x² y⁸ , for the ninth term.

Answer 2

(x+y)^n. kth term of a binomial expansion. r = k-1. C(n,r) x^(n-r)y^r

n = 10 k = 9 r = 8 x = x y = (-2y) C(10,8) x^2 (-2y)^8 = 45 x^2 *256 y 8 = 11520 x^2 y ^8