Computing the Euler Characteristic of the $n$-sphere

Question

Let $n\ge 2$. Compute the Euler characteristic of the $n$-sphere $S^n$ using the standard triangulation of the $n+1$-simplex.

I know the union of the proper faces of the $(n+1)$-simplex is homeomorphic to $S^n$, then also I am stuck...

Any help would be appreciated

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Answer 1

The number of $i$-dimensional faces of the $(n+1)$-simplex is ${n+2}\choose{i+1}$. Now, compute the sum of the terms $(-1)^{i}{{n+2}\choose{i+1}}$ for $i=0,1,\ldots,n$ and see what you get. (Hint: the answer depends on the parity of $n$.)