Solution: 2017 Fall Midterm - 9

Author: Michiel Smid

Question

What is the coefficient of $x^{20}y^{80}$ in the expansion of $(7x-13y)^{100}$?

(a)

$- {100 \choose 80} \cdot 7^{20} \cdot 13^{80}$

(b)

${100 \choose 20} \cdot 7^{80} \cdot 13^{20}$

(c)

${100 \choose 80} \cdot 7^{20} \cdot 13^{80}$

(d)

$- {100 \choose 20} \cdot 7^{80} \cdot 13^{20}$

$ = \sum^{100}_{k=0} \binom{100}{k} {(7x)}^{k} {(-13y)}^{100-k} $

$ = \sum^{100}_{k=0} \binom{100}{20} {(7x)}^{20} {(-13y)}^{100-20} $

$ = \binom{100}{20} {(7)}^{20} x^{20} {(-13)}^{80} y^{80} $

$ = \binom{100}{20} {(7)}^{20} {(-13)}^{80} x^{20} y^{80} $

$ = \binom{100}{20} {(7)}^{20} {(13)}^{80} x^{20} y^{80} $

Thus, the coefficient is $ \binom{100}{20} {(7)}^{20} {(13)}^{80} $