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A **permutation** is an arrangement, or listing, of objects in which the order is important. In the previous lessons, we looked at examples of the number of permutations of *n* things taken *n* at a time. Permutation is used when we are counting without replacement and the order matters. If the order does not matter then we can use combinations.

In these lessons, we will learn the permutation formula for the number of permutations of *n* things taken *r* at a time. We will also learn how to solve permutation word problems with repeated symbols and permutation word problems with restrictions.

Related Topics: More Lessons on Probability

In general P(*n*, *r*) means that the number of permutations of *n* things taken *r* at a time. We can either use reasoning to solve these types of permutation problems or we can use the permutation formula.

The formula for permutation is

If you are not familiar with the *n*! (*n* factorial notation) then have a look the factorial lesson.

**Example:**

A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different permutations of these letters can be made if no letter is used more than once?

*Solution: *

**Using reasoning:**

For the first letter, there are 5 possible choices. After that letter is chosen, there are 4 possible choices. Finally, there are 3 possible choices.

5 × 4 × 3 = 60

**Using the permutation formula:**

The problem involves 5 things (A, B, C, D, E) taken 3 at a time.

There are 60 different permutations for the license plate.

*Example: *

In how many ways can a president, a treasurer and a secretary be chosen from among 7 candidates?

*Solution: *

**Using reasoning:**

For the first position, there are 7 possible choices. After that candidate is chosen, there are 6 possible choices. Finally, there are 5 possible choices.

7 × 6 × 5 = 210

** Using permutation formula: **

The problem involves 7 candidates taken 3 at a time.

There are 210 possible ways to choose a president, a treasurer and a secretary be chosen from among 7 candidates

*Example: *

A zip code contains 5 digits. How many different zip codes can be made with the digits 0–9 if no digit is used more than once and the first digit is not 0?

*Solution: *

**Using reasoning:**

For the first position, there are 9 possible choices (since 0 is not allowed). After that number is chosen, there are 9 possible choices (since 0 is now allowed). Then, there are 8 possible choices, 7 possible choices and 6 possible choices.

9 × 9 × 8 × 7 × 6 = 27,216

** Using permutation formula: **

We can’t include the first digit in the formula because 0 is not allowed.

For the first position, there are 9 possible choices (since 0 is not allowed). For the next 4 positions, we are selecting from 9 digits.

The following video provides some information on permutations and how to solve some word problems using permutations.

In this video, we will learn how to evaluate factorials, use permutations to solve problems, determine the number of permutations with indistinguishable items.

Permutations - Counting Using Permutations.

Basic info on permutations and word problems using permutations are shown.

Introduction to permutations.

Permutations.

Permutations Involving Repeated Symbols - Example 1.

This video shows how to calculate the number of linear arrangements of the word MISSISSIPPI (letters of the same type are indistinguishable). It gives the general formula and then works out the exact answer for this problem.

Permutations Involving Repeated Symbols - Example 2.

In this example, we count how many 'stair-case' paths there are from the origin to the point (5,3).

1) In how many ways can five men and three women be arranged in a row if no two women is standing nest to one another?

2) In how many ways can the word "SUCCESS" be arranged if no two S's are next to on another?

Permutations with restrictions - letters/items stay together. 1) In how many ways can the letters in the word "HELLO" be arranged where the L's are together?

2) How many ways can the letters in the word 'PARALLEL" be arranged if the letters P and R are together?

This video highlights the differences between permutations and combinations and when to use each of them.

Learn how to differentiate between permutations and combinations, answer permutation questions using a calculator or the permutation equation, and how to answer combination questions using your calculator or the combination equation.