Solution: 2018 Fall Final - 6
Author: Michiel SmidQuestion
What is the coefficient of $x^{20}y^{35}$ in the expansion of $(5x - 3y)^{55}$?
(a)
${55 \choose 35} \cdot 5^{20} \cdot 3^{35}$
(b)
$- {55 \choose 35} \cdot 5^{20} \cdot 3^{35}$
(c)
${55 \choose 20} \cdot 5^{35} \cdot 3^{20}$
(d)
$- {55 \choose 20} \cdot 5^{35} \cdot 3^{20}$
Solution
$ = \sum_{k=0}^{55} \binom{55}{k} {(5x)}^{n-k} {(-3y)}^{k} $
We only consider $k=35$, as it results in $y^{35}$.
$ = \binom{55}{35} \cdot {(5x)}^{55-35} \cdot {(-3y)}^{35} $
$ = \binom{55}{35} \cdot 5^{20} \cdot {(-3)}^{35} \cdot x^{20} \cdot y^{35}$
$ = - \binom{55}{35} \cdot 5^{20} \cdot 3^{35} \cdot x^{20} \cdot y^{35}$
The coefficient is $ - \binom{55}{35} \cdot 5^{20} \cdot 3^{35} $