Solution: 2013 Fall Midterm - 8

Author: Pat Morin

Question

What is the coefficient of $ x^{12}y^{12} $ in the expansion of $ {(3x-7y)}^{24} $?

(a)

$-3^{12}7^{12}{{24}\choose{12}}$

(b)

$(3x)^{12}(7y)^{12}{{24}\choose{12}}$

(c)

$(3x)^{12}(-7y)^{12}{{24}\choose{12}}$

(d)

$21^{12}{{24}\choose{12}}$

$ = \sum^{24}_{k=0} \binom{24}{k} {(3x)}^k {(-7y)}^{24-k} $

$ = \binom{24}{12} {(3x)}^{12} {(-7y)}^{12} $

$ = \binom{24}{12} 3^{12} 7^{12} x^{12} y^{12} $

$ = \binom{24}{12} {21}^{12} x^{12} y^{12} $