A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, suppose we have a set of three letters: A, B, and C. we might ask how many ways we can arrange 2 letters from that set.

Permutation is defined and given by the following function:

## Formula

${^nP_r = \frac{n!}{(n-r)!} }$

Where −

- ${n}$ = of the set from which elements are permuted.
- ${r}$ = size of each permutation.
- ${n,r}$ are non negative integers.

### Example

**Problem Statement:**

A computer scientist is trying to discover the keyword for a financial account. If the keyword consists only of 10 lower case characters (e.g., 10 characters from among the set: a, b, c… w, x, y, z) and no character can be repeated, how many different unique arrangements of characters exist?

**Solution:**

Step 1: Determine whether the question pertains to permutations or combinations.

Since changing the order of the potential keywords (e.g., ajk vs. kja) would create a new possibility, this is a permutations problem.

Step 2: Determine n and r

n = 26 since the computer scientist is choosing from 26 possibilities (e.g., a, b, c… x, y, z).

r = 10 since the computer scientist is choosing 10 characters.

Step 2: Apply the formula

${^{26}P_{10} = \frac{26!}{(26-10)!} \\[7pt]

\ = \frac{26!}{16!} \\[7pt]

\ = \frac{26(25)(24)…(11)(10)(9)…(1)}{(16)(15)…(1)} \\[7pt]

\ = 26(25)(24)…(17) \\[7pt]

\ = 19275223968000 }$

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