Permutations P(n,r)

A permutation is an arrangement, or listing, of objects in which the order is important. In the previous lesson, we looked at examples of the number of permutations of n things taken n at a time. In this lesson, we will consider the number of permutations of n things taken r at a time.

Permutation Formula

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.

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