The coefficient of $a^{10}{b}^7{c}^3$  in the expansion of ${(bc+ca+ab)}^{10}$  is

 ${(ab+bc+ca)}^{10}={\left(abc\right)}^{10}{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})}^{10}$
$\Rightarrow$ Coeff of  $a^{10}{b}^7{c}^3{(ab+bc+ca)}^{10}$ 
=  co eff of $a^0{b}^{-3}{c}^{-7}$ in  ${(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})}^{10}=\frac{10!}{3!7!}=120$