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)
${80 \choose 100} 5^{20}3^{80}$
(c)
${100 \choose 80} 5^{80}3^{20}$
(d)
$-{100 \choose 80} 5^{20}3^{80}$
Solution
${(5x-36)}^{100}$
$=\sum^{100}_{k=0} \binom{100}{k}{(5x)}^k {(-3y)}^{n-k}$
$=\binom{100}{20}{(5x)}^{20} {(-3y)}^{80}$
$=\binom{100}{20}5^{20} 3^{80} x^{20} y^{80}$
$=\binom{100}{80}5^{20} 3^{80} x^{20} y^{80}$
$=\binom{100}{80}5^{20} 3^{80}$ (this is the coefficient)