Solution: 2015 Fall Midterm - 8
Author: Michiel SmidQuestion
What is the coefficient of $x^{81}y^{7}$ in the expansion of $(3x-17y)^{88}$?
(a)
$- {88 \choose 7} \cdot 3^7 \cdot 17^{81}$
(b)
${88 \choose 7} \cdot 3^7 \cdot 17^{81}$
(c)
$- {88 \choose 7} \cdot 3^{81} \cdot 17^7$
(d)
${88 \choose 7} \cdot 3^{81} \cdot 17^7$
Solution
$=\sum^{88}_{k=0} \binom{88}{k}{(3x)}^k {(-17y)}^{88-k}$
$=\binom{88}{7}{(3x)}^{81} {(-17y)}^{7}$
$=\binom{88}{7}3^{81} x^{81} {(-17)}^{7} y^{7}$
$=\binom{88}{7}3^{81} x^{81} {(-17)}^{7} y^{7}$
$=\binom{88}{7}3^{81} {(-17)}^{7} x^{81} y^{7}$
Multiplying a negative number by an odd number of negative numbers results in a negative number.
$=-\binom{88}{7}3^{81} {(17)}^{7} x^{81} y^{7}$