Find the number of solutions $(a_1, a_2, \dots, a_n)$ for the following equation:
$$a_1 + a_2 + \dots+ a_n = m$$
Where we are given that:
- $a_i \in \{x, x+1\}$ for $i=1,2,\dots,n$
- $x$ is some natural number
I know that if $x=0$, then $a_i \in \{0,1\}$, and then the solution is easy by having $m$ numbers from $(a_1, a_2, \dots, a_n)$ to be $1$ and the rest as $0$. But is that the only possible way of having a solution for the equation? If so, why then?