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)

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

(d)

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

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

We only consider $k=26$, as it results in $y^{26}$.

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

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

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

Thus, the coefficient is $ \binom{50}{26} \cdot 5^{24} \cdot 7^{26} $