Problem Find the number of integer solutions i.e. ordered pairs $\left(x_1, x_2, \ldots , x_r \right)$ such that $$x_1 + x_2 + \ldots + x_r = n, \quad x_i \geq 0, x_i \in \mathbb{Z}$$ Solution The number of such solutions is given by $${^{n+r-1}C_{r-1}} = \frac{\left( n+r-1 \right)!}{ n! \left( r-1 \right)!}$$