Combinatorics & probability

Combinatorics answers one question: how many ways? The two workhorses are permutations, which count ordered arrangements, and combinations, which count unordered selections. The number of ways to choose k items from n is C(n, k) = n! / (k!(n - k)!). Worked example: choosing 3 committee members from 10 people gives C(10, 3) = 10 x 9 x 8 / 6 = 120 ways, because the 3! = 6 orderings of the same three people are the same committee. Probability builds on counting: the probability of an event is the count of favourable outcomes divided by the count of all equally likely outcomes. The pigeonhole principle is the simplest counting argument of all: if more items than boxes exist, some box holds at least two items.