Question 2 Use induction to prove the statement $S(m, n)$ : "The number of possible (ordered) solutions to $x_{1}+x_{2}+$ $\cdots+x_{m}=n$ is $f(m, n)=\frac{(n+m-1) !}{n !(m-1) !}$ where $n$ and $m$ are positive integers and $x_{i} \in\{0\} \cup \mathbb{Z}^{+\cdots} .$ (Hint: First use induction to prove $S(m, 1)$ and $S(1, n)$. Proceed, again with induction, to prove $S(m+$ $1, n+1)$ using $S(m+1, n)$ and $S(m, n+1)$ as your basis steps. $)$