Which commonly known theorems have lesser known more beautiful versions?

Here is an example:

Kirchoff's matrix tree theorem says that the number of spanning trees of a graph can be computed by removing row i and column i from the associated Laplacian matrix L, and then taking the determinant. An alternative formulation is that the number of spanning trees is the product of the (n-1) nonzero eigenvalues of L divided by n, where n is the number of vertices.

The version that I find more beautiful states that the characteristic polynomial P(x) = det(L + xI) is the generating function of rooted spanning forests consisting of k trees.

The reason why I consider this version to be more beautiful is threefold:

1. The formula det(L + xI) is a little bit simpler than removing a row and column from L, and doesn't involve an arbitrary choice of i. 2. The formula det(L + xI) doesn't involve irrational quantities like eigenvalues. That's nice because the formula immediately suggests an algorithm to compute P(x) efficiently, unlike the formulation with the product of eigenvalues. 3. Most importantly, this version is more powerful because it tells you what all the coefficients of the characteristic polynomial count, not just what the coefficient of x\^1 counts. It therefore allows you to count spanning forests with any number of k components, not just spanning trees which have k=1 components. You can also count spanning forests with an unspecified number of components by computing P(1) = det(L + I).

Which theorems do you think have more beautiful versions?