The Euler Characteristic of Finite Subset Spaces

For a topological space $X$, the space of finite subsets $\mathrm{Sub}_n X$ consists of non-empty subsets of $X$ of cardinality at most $n$.

We compute the Euler characteristic of $\mathrm{Sub}_n X$ for any space $X$ having the homotopy type of a finite CW complex.

We obtain the explicit formula $\chi(\mathrm{Sub}_n X) = \sum_{i=1}^n \binom{\chi(X)}{i}$.