Question

Coefficient of $a^5{b}^4{c}^7$ in the expansion of $(bc+ca+ab{)}^8$ is _____

• A. 180
• B. 280
• C. 360
• D. 300

Answer 1

The correct option is B 280

General term of multinomial expansion is
$\frac{8!}{\left(r_1\right)!\left(r_2\right)!\left(r_3\right)!}\times {\left(\mathrm{bc}\right)}^{r_1}{\left(\mathrm{ca}\right)}^{r_2}{\left(\mathrm{ab}\right)}^{r_3}....(*)\mathrm{also}{r}_1+r_2+r_3=8....\left(1\right)\frac{8!}{\left(r_1\right)!\left(r_2\right)!\left(r_3\right)!}\times {\left(a\right)}^{r_2+r_3}{\left(b\right)}^{r_1+r_3}{\left(c\right)}^{r_1+r_2}\mathrm{coefficient}\mathrm{of}{a}^5{b}^4{c}^7\mathrm{is}\mathrm{required}\mathrm{hence},r_2+r_3=5...\left(2\right)r_1+r_3=4...\left(3\right)r_1+r_2=7...\left(4\right)\mathrm{substituting}\left(2\right)\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get}{r}_1=3\mathrm{substitute}{r}_1=3\mathrm{in}\left(3\right)\&\left(4\right),\mathrm{we}\mathrm{get}{r}_2=4r_3=1\mathrm{On}\mathrm{substituting}\mathrm{in}(*),\mathrm{we}\mathrm{get}-\mathrm{coefficient}\mathrm{is}\frac{8!}{3!4!1!}=\frac{8\times 7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!\times 1}=280$