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

The Correct Option is D

$ {{(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 $