Solution: 2019 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)

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

(d)

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

$(5x - 7y)^{50}$

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

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

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

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

Thus, the coefficient of $x^{24}y^{26}$ in the expansion of $(5x - 7y)^{50}$ is $\binom{50}{24} 5^{24} (7)^{26}$.