Binomial theorem
The binomial theorem gives a shortcut for expanding (x + y)^n without multiplying it out term by term: each term's coefficient comes from Pascal's triangle (or the combination formula n choose k), and the powers of x and y in each term always add up to n. For example, (x + y)^3 expands to x^3 + 3x^2y + 3xy^2 + y^3, where 1, 3, 3, 1 are the row-3 entries of Pascal's triangle: plugging in x=2, y=1 checks out both ways, since (2+1)^3 = 3^3 = 27 and 2^3 + 3(2)^2(1) + 3(2)(1)^2 + 1^3 = 8 + 12 + 6 + 1 = 27.
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