What Is Permutation?
A permutation is an arrangement, or listing, of objects inwhich the order is important. In previous lessons, we looked at examplesof the number of permutations of n things taken n at a time. Permutation is used when we arecounting without replacement and the order matters. If the order does not matter then we canuse combinations.
The following diagrams give the formulas for Permutation, Combination, and Permutation withRepeated Symbols. Scroll down the page with more examples and step by step solutions.
What Is The Permutation Formula?
In general P(n, r) means that the number of permutations of n things taken r at a time. We caneither use reasoning to solve these types of permutation problems or we can use the permutationformula.
The formula for permutation is
If you are not familiar with the n! (n factorial notation) then have a look thefactorial lessons.
Example:
A license plate begins with three letters. If the possible letters are A, B, C, D and E, howmany 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 4possible 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.
How To Use The Permutation Formula To Solve Word Problems?
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 are6 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 digits0–9 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 numberis chosen, there are 9 possible choices (since 0 is now allowed). Then, there are 8 possiblechoices, 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 4positions, we are selecting from 9 digits.
How To Solve Permutation Word Problems?
The following videos provide some information on permutations and how to solve some wordproblems using permutations.
In this video, we will learn how to evaluate factorials, use the permutation formula to solveproblems, 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?
Permutations With Indistinguishable Items
The number of different permutations of n objects where there are n1indistinguishable items, n2 indistinguishable items, … nkindistinguishable items, is n!/(n1!n2!…nk!).
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 ofthe same color look identical.
- Show Video Lesson
How To Calculate Permutations With Repeated Symbols?
Example:
How to calculate the number of linear arrangements of the word MISSISSIPPI (letters of the sametype are indistinguishable)?
Give the general formula and then work out the exact answer for this problem.
- Show Video Lesson
Permutations Involving Repeated Symbols
Example:
Count how many ‘stair-case’ paths there are from the origin to the point (5,3).
- Show Video Lesson
Determine The Number Of Permutations With Repeated Items
Example:
Find the number of distinguishable permutations of the given letters “AAABBC”
- Show Video Lesson
Determine The Number Of Permutations With Repeated Items
Example:
Find the number of distinguishable permutations of the given letters “AAABBBCDD”
- Show Video Lesson
How To Calculate Permutations With Restrictions Or Special Conditions?
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?
- Show Video Lesson
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?
- Show Video Lesson
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 mustbe on each of the ends assuming that each book looks different except for colour?
- Show Video Lesson
Compare Permutations And Combinations
This video highlights the differences between permutations and combinations and when to use each.
- Show Video Lesson
Try the free Mathway calculator andproblem solver below to practice various math topics. Try the given examples, or type in your ownproblem and check your answer with the step-by-step explanations.
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