Combinatorics and Newton’s binomial

The main question of combinatorics is “in how many ways?”. When counting options we list them in order, so none is missed or repeated. Addition rule: if A can be chosen in m ways or B in another n ways, then “A or B” can be chosen in m + n ways. Multiplication rule: if A can be chosen in m ways and then B in n ways, “A and B” can be done in mn ways; for consecutive steps the numbers of ways are multiplied. The number of arrangements of k out of n elements without repetition is Aₙᵏ = n(n − 1)…(n − k + 1); with repetition it is n^k. The number of ways to order all n elements (permutations) is n! = 1 · 2 · … · n, with 0! = 1. An n-element set has 2ⁿ subsets, since each element is either in the subset or not.