Number of positive integer solutions to the equation $a_1+a_2+\cdots+a_n=k$, where $k\geq n$

Question

I was wondering if there is a formula for counting the number of positive integers solutions to the equation
\begin{equation}
a_1+a_2+\cdots+a_n=k,
\end{equation}
where $n\leq k$, $k\in \mathbb{N}$ and $n\geq 2$. The case when $n>k$ is impossible since the least positive value that can assume all the $a_i$ are equal to $1$, and adding these we get $n=k$, a contradiction, so it does not have any solutions, but what about the case when $k\geq n$? For example, I wish to find a general formula for counting the positive integers solutions to equations such as $a+b=5$ and $a+b+c+d=12$, that is, where the number of variables is less than the constant positive integer. Any help or hints given will be greatly appreciated.

Answer 1

Imagine we have placed $k$ balls in line, $$\text{ooooooo}\quad (k=7).$$ Positive-integer solutions of $a_1+\cdots+a_n=k$ correspond to partitions of the balls into $n$ parts. For example, $$\text{oo}|\text{o}|\text{oooo}\quad (n=3,k=7)$$ corresponds to the solution $(2,1,4)$ of the equation $a_1+a_2+a_3=7$. We can choose $n-1$ places to put $n-1$ walls, among the $k-1$ midpoints of two balls next to each other. Therefore, the number of solution is the binomial coefficient $$\begin{pmatrix}k-1\\n-1\end{pmatrix}.$$