Mathematics
Combinatorial Identities and the Binomial Theorem
Quick fact
The binomial theorem shows that (1+x)^n expands into a sum where the k-th coefficient is the binomial coefficient C(n,k), which counts the ways to choose k items from n. This theorem also reveals that the sum of all coefficients in (1+x)^n equals 2^n, the number of subsets of an n-element set.
Why this is interesting
Ever notice that (x+y)^2 = x^2 + 2xy + y^2, and those coefficients (1,2,1) just happen to match the rows of Pascal's triangle? What if that pattern doesn't stop at 2?