What is the coefficient of $x^{20}y^{80}$ in the expansion of $(5x - 3y)^{100}$?
a) $-{100 \choose 80} 5^{20}3^{80}$
b) ${100 \choose 80} 5^{20}3^{80}$
c) ${100 \choose 80} 5^{80}3^{20}$
d) ${80 \choose 100} 5^{20}3^{80}$
Solution: 2014 Fall Midterm - 7
Author: Michiel SmidQuestion
What is the coefficient of $x^{20}y^{80}$ in the expansion of $(5x - 3y)^{100}$?
(a)
${100 \choose 80} 5^{20}3^{80}$
(b)
${100 \choose 80} 5^{80}3^{20}$
(c)
${80 \choose 100} 5^{20}3^{80}$
(d)
$-{100 \choose 80} 5^{20}3^{80}$
Solution
${(5x-36)}^{100}$
$ = \sum^{100}_{k=0} \binom{100}{k} {(5x)}^{n-k} {(-3y)}^{k} $
We only consider $k=80$, as it results in $y^{80}$.
$ = \binom{100}{80} \cdot {(5x)}^{100-80} \cdot {(-3y)}^{80} $
$ = \binom{100}{80} \cdot 5^{20} \cdot {(-3)}^{80} \cdot x^{20} \cdot y^{80} $
$ = \binom{100}{80} \cdot 5^{20} \cdot 3^{80} $ (final answer, i.e. the coefficient of $x^{20} y^{80}$)