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