Homotopy-theoretic computation of Euler characteristic of skeleton of simplex

Question

The Euler Characteristic of the $k$th skeleton of the $n$-simplex is $$\chi(\mathrm{sk}_k \Delta^n) = \binom{n + 1}{1} - \binom{n + 1}{2} + \cdots + (-1)^{k + 1}\binom{n + 1}{k + 1} = 1 + (-1)^k\binom{n}{k + 1}.$$ Is there a homotopy-theoretic and/or homology-theoretic proof of this fact?