In this exercise, I want you to pay attention to the coefficients.

Take the polynomial

Now, multiply it by 2 + x.

Now, multiply it by 2 + x.

Are you noticing a pattern, yet? Let's try once more.

Okay, there's no way this is a coincidence. You can find the number of subfacets in a hypercube of a certain dimensionality by looking at the binomial expansion of . For example, a tesseract has 16 vertices, 32 edges, 24 faces, 8 cells, and 1 teron. These are the coefficients of the last polynomial in that list above.

Neat applications of this are that you can use the binomial formula to quickly find the hypervolumes of *any* hypercube.

Just for fun, you could try non-integer exponents as well, like or , then use an extension of the binomial theorem to get the coefficients. Share what happens in the comments!

**Update**: You can use the binomial expansion if you rewrite the equation as . That might come in handy, since it allows for any

**Update 2**: You can use the Taylor Series expansion without rewriting the equation. This lets you have the exponent be any function of X, including such fun things as (log(X) + 2) dimensional hypercubes and stuff!