Question #a6ac5

1 Answer
Jan 26, 2018

-101376

Explanation:

Looking for the coefficient of a^5b^7 in the expansion of (a-2b)^12.

The binomial theorem says (x+y)^n=sum_(k=0)^n((n),(k))x^(n-k)*y^k

So (a-2b)^12 = sum_(k=0)^12((12),(k))a^(12-k)*(-2b)^k.

For the term we're seeking, we need the term when k=7:

((12),(7))a^(12-7)*(-2b)^7

=792a^5(-2)^7b^7

=-792*128a^5b^7

=-101376a^5b^7

so the coefficient is -101376.

Note:
((12),(7))=(12!)/(7!(12-7)!)
= ((12)(11)(10)(9)(8))/((5)(4)(3)(2)(1))
= (11)(9)(8) =792