How many solutions does the following Diophantine equation ($\forall \, i$, $x_i \in \mathbb{N}$) have:
$$\prod_{i = 1}^n {x_i} = k$$
where $n$ and $k$ are natural numbers?
Assume you know the prime factorisation of $k$:
$$k = \prod p_i^{\alpha_i}$$
where $p_i$ is prime and $\alpha_i$ is a natural number for all $i$.
For instance, when $k$ is the power of a prime ($k = p^t$ where p is prime and $t \in \mathbb{N}$), the problem reduces to a simple application of Stars and Bars. We want to divide the $t$ $p$'s into $n$ groups. This can be done in $\binom{n + t - 1}{n}$ ways.
Each prime can be divided separately using this property; the number of ways is:
$$\prod_{i = 1}^r {\binom{n + \alpha_i - 1}{n}}$$
if there are $r$ prime factors.