Permutations

Once objects can be told apart and their order matters, arranging them is called a permutation. This is the key word to watch for in a word problem: if changing the order of the same objects counts as a different outcome (seating order, a rank list, a password, a number formed from digits), it is a permutation problem.

Definition and formula

A permutation is an arrangement of a set of objects in a definite order. The number of permutations of $n$ distinct objects taken $r$ at a time (where $0 \le r \le n$) is denoted $^{n}P_{r}$ or $P(n,r)$, and is given by

Derivation using the multiplication principle. To fill $r$ positions in order using $n$ distinct objects: the first position can be filled in $n$ ways; once used, the second position can be filled in $(n-1)$ ways; the third in $(n-2)$ ways; and so on, down to the $r$-th position, which can be filled in $n-(r-1) = n-r+1$ ways. Multiplying these:

Special cases

• All $n$ objects taken all at a time: $^{n}P_{n} = \dfrac{n!}{0!} = n!$
• $r = 0$: $^{n}P_{0} = \dfrac{n!}{n!} = 1$ (there is exactly one way to arrange nothing — the empty arrangement).
• $^{n}P_{1} = n$ — choosing and "arranging" just one object out of $n$ is simply choosing one of the $n$ objects.

Permutations when some objects repeat

If $n$ objects are to be arranged, but among them $p$ objects are of one kind (identical to each other), $q$ of a second kind, $r$ of a third kind, and so on, then swapping identical objects among themselves does not create a genuinely new arrangement. The number of distinct arrangements is therefore reduced by dividing out the repeated factorials:

This correction is essential whenever a word problem involves arranging the letters of a word that has repeated letters, or arranging beads/objects where some are indistinguishable.

Circular permutations (brief note) …