Mathematics - Permutation & Combination Question with Solution | TestHub

Let $x,y,z$ be the number of sticks received by children A, B, and C. The line segments formed by them form a triangle if and only if $1\le x\le 5$, $1\le y\le 5$, and $1\le z\le 5$.

Therefore, $n$ is the number of positive integral solutions of the equation $x+y+z=12$, where $x,y,z\le 5$.

$$n=Coefficientof{x}^{12}in(x+\dots +x^5{)}^3$$

$$n=Coefficientof{x}^{9}in(1+x+x^2+x^3+x^4{)}^3$$

$$n=Coefficientof{x}^{9}in{\left(\frac{1-x^5}{1-x}\right)}^3$$

$$n=Coefficientof{x}^{9}in(1-x^5{)}^3(1-x{)}^{-3}$$

$$n=Coefficientof{x}^{9}in(1-3x^5+3x^{10}-x^{15})(1-x{)}^{-3}$$

$$n=1\times {}^{11}{C}_2-3\times {}^6{C}_2$$

$$n=55-45=10$$