$Y\_1, Y\_2, Y\_3,...,Y\_n$ are independent and identically distributed random variables and they have uniform distribution on ${\{1,...,m\}}$. 

Prove that for $n\le k\le mn$, we have

$$P(Y\_1+Y\_2+Y\_3+...+Y\_n=k)=\frac 1 {m^n} \sum\_{0\le i \le(k-n)/m} (-1)^i \binom{n}{i} \binom{k-mi-1}{n-1}$$

So I think this is related to the binomial formula with $$\{1+x\}^a=\sum\_{i=0}^{\infty}\binom{a}{i}x^i$$ where $|a| \lt 1$. How would I transform this formula to the given identity in the question?