  Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for
counting the number of monic polynomials that are of degree $n$ and have
exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also
compare our results with the analogous existing ones in the integer case, where
one studies all the natural numbers up to $x$ with exactly $k$ prime factors.
In particular, we show that the number of monic polynomials grows at a
surprisingly higher rate when $k$ is a little larger than $\log n$ than what
one would speculate from looking at the integer case.
