binomial theorem
The binomial theorem is a formula for expanding a binomial raised to a power, like (a + b)^n, without multiplying it out factor by factor. Instead of slogging through (a + b)·(a + b)·...·(a + b), it tells you the whole answer at a glance.
It says (a + b)^n equals the sum, for k from 0 to n, of C(n, k)·a^(n−k)·b^k, where C(n, k) is the binomial coefficient “n choose k.” As you read across the expansion, the power of a steps down from n to 0 while the power of b steps up from 0 to n, and their exponents always add to n.
The coefficients C(n, k) are exactly the numbers in row n of Pascal's triangle, which is why the triangle and the theorem are two views of the same fact. With a minus sign, (a − b)^n is the same expansion with alternating signs, since you are raising (a + (−b)) to the power.
(a + b)^3 = a^3 + 3a^2 b + 3a b^2 + b^3, with coefficients 1, 3, 3, 1 from row 3 of Pascal's triangle.
Exponents of a fall while those of b rise; each pair adds to 3.