This is a bit of cheating (since requires to know a bit of homological algebra) but is too nice to not be mentioned:
Let's notice that the generating function for dimensions of graded components for the exterior algebra on elements of degrees $a_1,\ldots,a_k,\ldots$ is $\prod (1+t^{a_i})$;
Let's recall that for a Lie algebra $\mathfrak{g}$ the exterior algebra $\Lambda^\bullet(\mathfrak{g})$ is equipped with a Chevalley--Eilenberg differential, making it into a chain complex. If $\mathfrak{g}$ is graded with basis elements of degrees $a_1,\ldots,a_k,\ldots$, the generating function of Euler characteristics of graded components of this chain complex is obtained from the previous formula by incorporating signs, amounting to $\prod (1-t^{a_i})$;
Of course, if we know the homology of the Chevalley--Eilenberg complex, we can compute that generating function in a different way, and get a combinatorial identity. Some famous identities arise that way for infinite-dimensional Lie algebras. For instance, one can prove Euler's pentagonal theorem considering $\mathfrak{g}=L_1(1)$, the algebra of vector fields on the line that vanish twice at zero. More generally, for nilpotent subalgebras of Kac--Moody algebras one obtains Macdonald's identities, see e.g. "Lie algebra homology and the Macdonald-Kac formulas" by Garland and Lepowsky.
