  We discuss a product formula for $F$-polynomials in cluster algebras, and
provide two proofs. One proof is inductive and uses only the mutation rule for
$F$-polynomials. The other is based on the Fock-Goncharov decomposition of
mutations. We conclude by expanding this product formula as a sum and
illustrate applications. This expansion provides an explicit combinatorial
computation of $F$-polynomials in a given seed that depends only on the
$\mathbf{c}$-vectors and $\mathbf{g}$-vectors along a finite sequence of
mutations from the initial seed to the given seed.
