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

In this case, we write 

$\begin{array}{r}{\mathrm{a}}^3{\mathrm{b}}^4{\mathrm{c}}^5=\left(\mathrm{ab}{)}^{\mathrm{x}}\right(\mathrm{bc}{)}^{\mathrm{y}}(\mathrm{ca}{)}^{\mathrm{z}}\\ \Rightarrow {\mathrm{a}}^3{\mathrm{b}}^4{\mathrm{c}}^5={\mathrm{a}}^{\mathrm{z}+\mathrm{x}}\cdot {\mathrm{b}}^{\mathrm{x}+\mathrm{y}}\cdot {\mathrm{c}}^{\mathrm{y}+\mathrm{z}}\\ \Rightarrow \mathrm{z}+\mathrm{x}=3,\mathrm{x}+\mathrm{y}=4,\mathrm{y}+\mathrm{z}=5 .....\left(1\right)\end{array}$

On adding these equations, we get 

$\therefore \begin{array}{r}2(\mathrm{x}+\mathrm{y}+\mathrm{z})=12\\ \mathrm{x}+\mathrm{y}+\mathrm{z}=6\end{array}.......\left(2\right)$

On solving Eqs. (i) and (ii), we get

x=1,Y =3 ,Z =2 

Therefore, the coefficient of ${\mathrm{a}}^3{\mathrm{b}}^4{\mathrm{c}}^5$in the expansion of
(bc+ ca+ ab)⁶
The coefficient of (ab)¹(bc)³(ca)² in the expansion of 

(bc+ca+ab)⁶ 

$=\frac{6!}{1!3!2!}=60$