28. The Many Faces of the n-simplex

Let $C_n$ be an $n$-dimensional cube.

1. For each integer $k$ with $0\leq k\leq n$, derive a general formula for the number of $k$-dimensional faces of $C_n$.

2. Prove your formula rigorously, starting from the structure of an $n$-dimensional cube rather than assuming a general formula.

3. Let $S_n$ be an $n$-dimensional simplex. Derive and prove a corresponding formula for the number of $k$-dimensional faces of $S_n$, for every $0\leq k\leq n$.

4. Compare the resulting formulas for the $n$-cube and the $n$-simplex, and explain the combinatorial reason for the difference between them.