The Euler Characteristic of Finite Subset Spaces

For a topological space $X$, the space of finite subsets ${\mathrm{S}\mathrm{u}\mathrm{b}}_{n}X$ consists of non-empty subsets of $X$ of cardinality at most $n$. We compute the Euler characteristic of ${\mathrm{S}\mathrm{u}\mathrm{b}}_{n}X$ for any space $X$ having the homotopy type of a finite CW complex. We obtain the explicit formula $\chi ({\mathrm{S}\mathrm{u}\mathrm{b}}_{n}X)={\sum }_{i=1}^n\left(\genfrac{}{}{0px}{}{\chi (X)}{i}\right)$.