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 or special conditions.

**Related Pages**

Permutations

Permutations and Combinations

Counting Methods

Factorial Lessons

Probability

A **permutation** is an arrangement, or listing, of objects in
which the order is important. In 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.

The following diagrams give the formulas for Permutation, Combination, and Permutation with Repeated Symbols. Scroll down the page with more examples and step by step solutions.

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 lessons.

**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 the 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 the 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 videos provide some information on permutations and how to solve some word problems using permutations.

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

A permutation is an arrangement or ordering. For a permutation, the order matters.

**Example:**

How many different ways can 3 students line up to purchase a new textbook reader?

**Solution:**

n-factorial gives the number of permutations of n items.

n! = n(n - 1)(n - 2)(n - 3) … (3)(2)(1)

Permutations of n items taken r at a time.

P(n,r) represents the number of permutations of n items r at a time.

P(n,r) = n!/(n - r)!

**Examples:**

- Find P(7,3) and P(15,5)
- If a class has 28 students, how many different arrangements can 5 students give a presentation to the class?
- How many ways can the letters of the word PHOENIX be arranged?

The number of different permutations of n objects where there are n_{1}
indistinguishable items, n_{2} indistinguishable items, … n_{k}
indistinguishable items, is n!/(n_{1}!n_{2}!…n_{k}!).

**Examples:**

- How many ways can the letters of the word MATHEMATICS be arranged?
- How many ways can you order 2 blue marbles, 4 red marbles and 5 green marbles? Marbles of the same color look identical.

**Example:**

How to calculate the number of linear arrangements of the word MISSISSIPPI (letters of the same
type are indistinguishable)?

Give the general formula and then work out the exact answer for this problem.

**Example:**

Count how many ‘stair-case’ paths there are from the origin to the point (5,3).

**Example:**

Find the number of distinguishable permutations of the given letters “AAABBC”

**Example:**

Find the number of distinguishable permutations of the given letters “AAABBBCDD”

**Permutations with restrictions: items not together.**

**Example:**

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

**Example:**

- In how many ways can the letters in the word “HELLO” be arranged where the L’s are together?
- How many ways can the letters in the word ‘PARALLEL" be arranged if the letters P and R are together?

**Permutations with restrictions: items are restricted to the ends**

**Example:**

- In how many ways can 2 men and 3 women sit in a line if the men must sit on the ends?
- In how many ways can 3 blue books and 4 red books be arranged on a shelf if a red book must be on each of the ends assuming that each book looks different except for colour?

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

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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