Question

Find coefficient of $a^3{b}^4{c}^5$ in the expansion of ${(bc+ca+ab)}^6$

Answer 1

We write $a^3{b}^4{c}^5={\left(ab\right)}^x{\left(bc\right)}^y{\left(ca\right)}^z$

$\therefore a^3{b}^4{c}^5=a^{z+x}{b}^{x+y}{c}^{y+z}$

$\Rightarrow z+x=3,x+y=4,y+z=5$

On adding all we get

$z+x+x+y+y+z=2(x+y+z)=3+4+5=12$

$\Rightarrow x+y+z=\frac{12}{2}=6$

Then $x=1,y=3,z=2$

$\therefore$ the coefficient of $a^3{b}^4{c}^5$ in the expansion of ${(bc+ca+ab)}^6$

or the coefficient of ${\left(ab\right)}^1{\left(bc\right)}^3{\left(ca\right)}^2$ in the expansion of ${(bc+ca+ab)}^6$ is $\frac{6!}{1!3!2!}=\frac{6\times 5\times 4\times 3!}{3!\times 2}=60$