Let us say we have an n-dimensional hypercube. How many k-dimensional hypercubes does it contain?
Some observations:
Let us call have a function $S_k(n)$ which returns how many k-cubes are in a n-cube. $$k>n \implies S_k(n)=0$$ $$k=n \implies S_k(n)=1$$ $$k=n-1 \implies S_k(n)=2n$$ $$S_0(n)=2^n$$
