K-dimensional faces of an N-dimensional hypercube

"Exploded" Cube

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:

"Exploded" 1-D cube

Now, the two-dimensional case:

"Exploded" Square

The 3-dimensional case, pictured at the top of this blog, illustrates (1 + x + 1)^3.

Get it?

That is, the number of K-dimensional faces of the N-dimensional hypercube is the coefficient of x^K in the expansion of (1 + x+ 1)^N, which, using the binomial theorem, is equal to {N \choose K}2^{N-K}.


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