Permutations: Arrangements in Order

A permutation is an arrangement of objects in a definite order. Order matters: the arrangement $AB$ is counted as different from $BA$.

Suppose $r$ objects are to be arranged in a row, chosen from $n$ distinct objects (no object used twice). The first place can be filled in $n$ ways; the second in $(n-1)$ ways (one object already used); the third in $(n-2)$ ways; and so on, down to the $r$-th place in $(n-r+1)$ ways. By the multiplication principle:

Example: the number of ways to arrange $3$ of the letters $A, B, C, D, E$ in a row is $^{5}P_{3} = \dfrac{5!}{2!} = \dfrac{120}{2} = 60$, i.e. $5 \times 4 \times 3 = 60$.