Related Pages
Permutations
Permutations and Combinations
Counting Methods
Factorial Lessons
Probability
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. Many examples are given together with answers.
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:
The number of different permutations of n objects where there are n1 indistinguishable items, n2 indistinguishable items, … nk indistinguishable items, is n!/(n1!n2!…nk!).
Examples:
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:
Permutations with restrictions: letters/items stay together
Example:
Permutations with restrictions: items are restricted to the ends
Example:
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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