# Permutations

Definition: A permutation is an ordered combination of a set of items.

For example, ABC is a distinct permutation from ACB because they are ordered differently. (When order doesn’t matter, it’s called a combination.)

To calculate a possible number of permutations, we use the formula for nPr, where n represents the total number of items being chosen from, and r represents the number of items we are choosing (“n different things taken r at a time”):

$\dpi{150} \bg_white \large n\,P\,r = \frac{n!}{(n-r)!}$

Recall the ! symbol means factorial.

When using permutation notation, nPr, we read it as “n choose r”.

Example: A professor has ten books and wants to display five of them on a bookshelf. How many different ways can he arrange the books?

Solution:

Step 1: First we must decide if the order in this situation matters. Because the books are being arranged in a particular way on a shelf, the order will be significant. Therefore we’re being asked to identify the number of permutations.

Step 2: Identify the n and r for this question. Since we are choosing from 10 total books, our n is 10. Since we are choosing the books 5 at a time, our r is 5. (“10 different things taken 5 at a time.”)

Step 3: Plug the values into the equation and simplify to determine how many combinations there are
$\dpi{150} \bg_white \\n\,P\,r = \frac{n!}{(n-r)!}\\\\ 10\,\,\textup{P}\,\,5 = \frac{10!}{(10-5)!}\\\\ {\;\;\;\;\;\;\;\;\;\;}= \frac{10\,\cdot\,9\,\cdot\,8\,\cdot\,7\,\cdot\,6\,\cdot\,5\,\cdot\,4\,\cdot\,3\,\cdot\,2\,\cdot\,1\,\cdot\, }{5\,\cdot\,4\,\cdot\,3\,\cdot\,2\,\cdot\,1}\\\\ {\;\;\;\;\;\;\;\;\;\;}= \frac{3628800}{120} = {\color{DarkGreen} 30240}$

This tells us there are 30240 different ways in which the professor can arrange his 10 books into groups of 5.