Question

The coefficient of $a^8{b}^4{c}^9{d}^9$ in $(abc+abd+acd+bcd{)}^{10}$ is

• A. $10!$
• B. $\frac{10!}{8!4!9!9!}$
• C. $2520$
• D. none of these

Answer 1

Answer: (C)

$a^{10}{b}^{10}{c}^{10}{d}^{10}{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)}^{10}$
Therefore the required coefficient is equal to the coefficient of
$a^{-2}{b}^{-6}{c}^{-1}{d}^{-1}$ in ${\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)}^{10}$,
which is given by
$\frac{10!}{2!6!1!1!}=\frac{10\times 9\times 8\times 7}{2}$
$=2520$