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

Answer 1

Answer: (D)

1260

Answer 2

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