How many K-dimensional faces does an N-dimensional hypercube have? The picture above shows that when N = 3, there are eight 0-dimensional faces (vertices), twelve 1-dimensional faces (edges), six 2-dimensional faces, and one 3-dimensional “face.” But what’s the answer in general?
Here’s a way to “see” it.
First, the one-dimensional case:
Now, the two-dimensional case:
The 3-dimensional case, pictured at the top of this blog, illustrates .
That is, the number of K-dimensional faces of the N-dimensional hypercube is the coefficient of in the expansion of , which, using the binomial theorem, is equal to .