Mathematics - Permutation & Combination Question with Solution | TestHub

Let ${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3$ and ${\mathrm{x}}_4$ be the number of white, red, blue and green balls that are selected. Then ${\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3+{\mathrm{x}}_4=10$. The required number of ways

= coefficient of $y^{10}$ in ${\left(1+y+y^2+y^3+\dots \right)}^4$

= coefficient of $y^{10}$ in $(1-y{)}^{-4}$

= coefficient of $y^{10}$ in ( $1+{}^4{C}_{1}y+{}^5{C}_2{y}^2+{}^6{C}_3{y}^3+\dots$.)

$={}^{13}{\mathrm{C}}_3=\frac{13\times 12\times 11}{2\times 3}=286$