Permutation

An arrangment of the order of a Set

A permutation is an Isomorphism of a set

Partial Permutation

$$P(n,k)=P_{n,k}=\frac{n!}{(n-k)!}=C(n,k)\cdot P(k,k)$$

Repetitious Permutation

$$n^k$$

Multiset Permutation

a.k.a Multinomial Coefficients

The formula to find the number of distinct arrangements of a set of objects, some of which are indistinguishable

• where $n$ is the count of items
• where each $k_i$ is the number of repetitions of an object

$$\frac{n!}{k_1!\cdot k_2!\cdot \cdots \cdot k_n!}=\frac{\text{Orderings ignoring repeats}}{\text{Product of orderings of each repeated item}}$$

For example, How many distinct arrangements can be made using the letters in the word: BANANA?

Solution

• Pretend like there are no duplicate letters: $6!$ arrangements
• Of those arrangements some will be exactly the same Consider BANANA. We want to elimate the repeated arrangements How many ways can you arrange 3 As? $3!$ How many ways can you arrange 2 Ns? $2!$

$$\frac{6!}{3!2!}=60$$