Permutations — nPr, Arrangements with Repetition, and Circular Permutations

A permutation is an arrangement of objects in a definite order — changing the order gives a different permutation.

Permutations of $n$ distinct things taken $r$ at a time, written $^{n}P_{r}$, counts the number of ways to arrange $r$ objects chosen from $n$ distinct objects, order mattering:

Special cases: $^{n}P_{n} = n!$ (arranging all $n$ objects) and $^{n}P_{0} = 1$ (there is exactly one way to arrange nothing).

Permutations when some things are alike (repetition). If $n$ objects are to be arranged but $p_1$ of them are alike of one kind, $p_2$ alike of a second kind, and so on, the number of distinct arrangements is

This divides out the arrangements that look identical because some objects cannot be told apart.

Circular permutations. Arranging $n$ distinct objects around a circle is different from arranging them in a row, because a circular arrangement has no fixed starting point — rotating everyone by one seat gives what looks like the same arrangement. The number of distinct ways to seat $n$ distinct people around a round table is

(one person's position is fixed as a reference, and the remaining $n-1$ people are arranged relative to that position). If, in addition, a clockwise arrangement is considered identical to its mirror-image anticlockwise arrangement (as with beads on a necklace, where the object can be flipped over), the count halves to $\dfrac{(n-1)!}{2}$. …