Permutation

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A permutation of a set of $r$ objects is any rearrangement (linear ordering) of the $r$ objects. There are $\displaystyle r!$ (the factorial of $r$) permutations of a set with $r$ objects.


An important question is how many ways to pick an $r$-element subset of a set with $n$ elements, where order matters. To find how many ways we can do this, note that for the first of the $r$ elements, we have $n$ different objects we can choose from. For the second element, there are $n-1$ objects we can choose, $n-2$ for the third, and so on. In general, the number of ways to permute $r$ objects from a set of $n$ is given by $P(n,r)=n(n-1)(n-2)\cdots(n-r+1)=\frac{n!}{(n-r)!}$.

See also