Permutations

Permutation without Repetition

A permutation is an arrangement, or listing, of n distinct objects in which the order is important.

Total number of permutations in case of n elements:

P_{n} = n \cdot (n - 1) \cdot (n - 2) \cdot . . . \cdot 2 \cdot 1 = n!

Example:

In case of 4 elemts: {a,b,c,d}: $n = 4, P_{4} = 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$

A permutation is an arrangement, or listing, of n objects in which the order is important. The elements are repeated. Number of repetations:

k_{1}, k_{2}, k_{3}, . . ., k_{r}; (k_{1} + k_{2} + k_{3} + . . . + k_{r} \leq n)

Total number of permutations:

P_{n}^{k_{1}, k_{2}, k_{3}, . . ., k_{r}} = \frac{n!}{k_{1}! \cdot k_{2}! \cdot k_{3}! \cdot . . . \cdot k_{r}!}

Example:

In case of 7 elemet: {a,a,a,a,b,b,c} first element repeats 4 times, second element repeats 2 times: $n = 7, k_{1} = 4, k_{2} = 2, k_{1} = 1$

Total number of permutations:

P_{7}^{4, 2, 1} = \frac{7!}{4! \cdot 2! \cdot 1!} = 105

Keywords: Permutation without Repetition with Repetition