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”):

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

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