Solution: 2016 Fall Midterm - 9

Author: Michiel Smid

Question

What is the coefficient of $x^{24}y^{26}$ in the expansion of $(5x - 7y)^{50}$?

(a)

$- {50 \choose 26} \cdot 5^{24} \cdot 7^{26}$

(b)

${50 \choose 24} \cdot 5^{26} \cdot 7^{24}$

(c)

$- {50 \choose 24} \cdot 5^{26} \cdot 7^{24}$

(d)

${50 \choose 26} \cdot 5^{24} \cdot 7^{26}$

$ = \sum^{50}_{k=0} \binom{50}{k} {(5x)}^{50-k} {(-7y)}^k $

$ = \sum^{50}_{k=0} \binom{50}{26} {(5x)}^{50-26} {(-7y)}^k $

$ = \binom{50}{24} {(5)}^{24} x^{24} {(-7)}^{26} y^{26} $

$ = \binom{50}{26} {(5)}^{24} {(-7)}^{26} x^{24} y^{26} $