# Useful Formulae

22 COMBINATIONSPERMUTATIONS ways of arranging things Suppose we have nitems and we wish to consider groupings of k of them. Two types of groupings exist combinations, where the order is not important, and permutations, where the order does matter. We need to use factorials here. The factorial of m,written m where m 1 is defined as m mm1m2 2x1 e.g.6 6x5x4x3x2x1 720 0 is a special case 0 1 by convention. The number of combinations of kitems out of nis then C n k , also written The obviously larger number of permutations is P n k k In a situation having two possible outcomes, Pand Q, with probabilities p and q,we know p q 1,so p qn 1. The term pnkqk in the binomial expansionof pqn is then the probability of nkoccurrences of Pand koccurrences of Qfrom a total of n. The general binomial formula is xyn xn xn1y xnkyk xyn1 yn The rows of Pascals Triangle correspond here. For example xy4 x4 4x3y 6x2y2 4xy3 y4 n k n nk k k n n nk k n 1 n n n n1 n k n 23