combination

A combination is a sub-selection of elements from a given set of elements. Typically we are concerned to count the number of possible combinations, that is, the number of possible ways to select a given number of elements from a set of some given size.

The number of ways of selecting \(r\) elements from a set of \(n\) elements is generally denoted by \(\left(\begin{array}{c}n \\ r \end{array} \right)\) and described as ‘\(n\) choose \(r\).’ It is equal to the binomial coefficient and may be calculated according to the formula

\[\left(\begin{array}{c}n \\ r \end{array} \right) = \frac{n!}{r!(n-r)!}\]

It is also sometimes denoted by \(\mbox{}_nC_r\).