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 .

Get it?

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 .

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