The binomial theorem: expansion, general term and sums of expansions

A binomial is a sum of two terms, such as x + a. Multiply it by itself and a pattern appears: (x + a)² = x² + 2xa + a² and (x + a)³ = x³ + 3x²a + 3xa² + a³. In (x + a)ⁿ there are n + 1 terms; the powers of x fall from n to 0 while the powers of a rise from 0 to n, and in every term the two powers add up to n. The coefficients are the rows of Pascal’s triangle: each number is the sum of the two numbers above it. Instead of building the triangle row by row, we can name its entries. The entry in row n and place r (places are counted from 0) is the number of ways to choose r objects out of n, written nCr and read ‘n choose r’ (in Arabic التوافيق). Its value is n!/(r!(n − r)!), where n! = n × (n − 1) × … × 1 and 0! = 1. For quick work cancel: 8C3 = (8 × 7 × 6)/(3 × 2 × 1) = 56. Two useful facts: nC0 = nCn = 1 and nCr = nC(n − r). The numbers of row n add up to 2ⁿ, the number of all subsets of a set of n objects (each object is either in or out). The binomial theorem says (x + a)ⁿ = nC0 xⁿ + nC1 xⁿ⁻¹a + nC2 xⁿ⁻²a² + … + nCn aⁿ. For (x − a)ⁿ put −a in place of a, so the signs alternate: + − + − …