Question

The coefficient of $a^5{b}^6{c}^7$ in the expansion of ${(bc+ca+ab)}^9$ is

• A. 100
• B. 120
• C. 720
• D. 1260
• E. 1440

Answer 1

Answer: (D)

${(bc+ca+ab)}^9={[bc+a(b+c)]}^9$
$\therefore$ Coefficient of $a^5{b}^6{c}^7<br>$= coefficient of $a^5{b}^6{c}^7$ in ${}^9{C}_5$ in ${(bc)}^4{a}^5{(b+c)}^5<br>$= coefficient of $b^2{c}^3$ in ${}^9{C}_5{(b+c)}^5$ ${=}^9{C}_5{\times }^5{C}_2=\frac{9!}{5!\times 4!}\times \frac{5!}{3!\times 2!}$
$=\frac{9\times 8\times 7\times 6\times 5}{3\times 2\times 1\times 2}=1260$