Question

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

• A. $10!$
• B. $1260$
• C. $5040$
• D. $2520$

Answer 1

The correct option is D $2520$

$(abc+abd+acd+bcd{)}^{10}=a^{10}{b}^{10}{c}^{10}{d}^{10}{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})}^{10}=a^{10}{b}^{10}{c}^{10}{d}^{10}{(a^{-1}+b^{-1}+c^{-1}+d^{-1})}^{10}$

The coefficient of $a^8{b}^4{c}^9{d}^9$ is equal to the coefficient of $a^{-2}{b}^{-6}{c}^{-1}{d}^{-1}$ in
${(a^{-1}+b^{-1}+c^{-1}+d^{-1})}^{10}$

$=\frac{10!}{2!6!1!1!}=\frac{10!}{2!6!}=\frac{10\times 9\times 8\times 7}{2}=2520$