The basic rule of combinatorics

Combinatorics counts the number of different ways of choosing and arranging objects. The basic rule is the multiplication rule: if the first choice can be made in m ways and the next choice in n ways, both choices together can be made in m · n ways. For example, 3 kinds of shirts and 4 kinds of trousers give 3 · 4 = 12 outfits. When the choices are alternatives joined by “or”, the addition rule applies: the numbers of ways are added, for instance one drink out of 3 juices or 4 teas can be chosen in 3 + 4 = 7 ways. When forming numbers from digits, we note whether repetition is allowed: without repetition the number of options drops by one at each next place, with repetition it stays the same; the first digit cannot be 0. The number of pairs that can be formed from n objects is n(n - 1)/2, because the product n(n - 1) counts every pair twice.