Intuition behind negative combinations

I know two "explanations" of this phenomenon. One idea is to replace cardinality with a form of Euler characteristic; this is described, for example, in Propp's Exponentiation and Euler measure.

The other is to replace cardinality with dimension (of a vector space) and then come up with a reasonable notion of negative dimension. The starting observation is that if $V$ is a vector space of dimension $n$, then the exterior power $\Lambda^k(V)$ has dimension ${n \choose k}$, whereas the symmetric power $S^k(V)$ has dimension $(-1)^k {-n \choose k}$. It turns out one can think of the exterior power as being the symmetric power but applied to a "vector space of negative dimension." I briefly explain this story in this blog post.