Combinations

In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases it is possible to count the number of combinations. For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient

which can be written using factorials as ${\displaystyle \textstyle {\frac {n!}{k!(n-k)!}}}$ whenever ${\displaystyle k\leq n}$, and which is zero when ${\displaystyle k>n}$. The set of all k-combinations of a set S is sometimes denoted by ${\displaystyle \textstyle {\binom {S}{k}}\,}$.

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