How to Use the Permutation Calculator
This calculator helps you find permutations with or without repetition and shows the calculation step-by-step. Thus, with this free calculator, you can quickly find the number of ways you can arrange r items from n available items when the order matters.
To use the calculator:
- Enter the total number of available items, n.
- Enter the number of items you want to arrange, r.
- Leave the With repetition? toggle turned off for a standard permutation. If the same item can be used more than once, turn the toggle on.
- Click Calculate.
The calculator instantly returns the number of permutations and shows how the answer was calculated step-by-step.
Working with a problem where order does not matter? Use the combination calculator instead.
What Is a Permutation?
A permutation is an arrangement or selection of items in which order matters. For example, suppose Ava, Ben, and Chloe are competing for first and second place. If Ava finishes first and Ben finishes second, the result is different from Ben finishing first and Ava finishing second. Therefore, Ava, Ben ≠ Ben, Ava.
Because changing the order creates a different outcome, this is a permutation.
Permutation Formula
The permutation formula is: nPr = n!/(n-r)!
Where:
- n is the total number of available items;
- r is the number of items being arranged; and
- ! represents factorial.
The formula allows you to calculate the number of ways to pick and arrange r items from a set of n distinct items, where order matters and without repetition.
Note. For the permutation formula to work, 0 ≤ r ≤ n
However, if order matters and the same item can be used more than once (with repetition), you can find the permutation using nr.
Where:
- n is the number of available choices for each position; and
- r is the number of positions being filled.
Our permutation calculator supports this option. You only need to turn on the “with repetition?” toggler if the problem suggests that the same item appears more than once and order matters.
How to Calculate Permutations
To calculate a permutation manually:
- Identify the total number of available items (n).
- Identify the number of items being arranged (r).
- Determine whether repetition is allowed.
- Apply the appropriate permutation formula.
Example 1: Permutation Without Repetition
Nine students have reached the final stage of a public-speaking competition. Four students will receive first, second, third, and fourth place. How many different possible finishing orders are there?
Solution
To solve the problem manually, follow these steps.
Step 1: Identify the total number of available items (n).
There are 9 finalists. Therefore, n = 9
Step 2: Identify the number of items being arranged (r).
Four students will occupy the four ranked positions. Therefore, r = 4
Step 3: Determine whether repetition is allowed.
A student cannot occupy more than one finishing position. Therefore, repetition is not allowed. Also, the order matters because finishing first is different from finishing second, third, or fourth.
Step 4: Apply the appropriate permutation formula.
Since order matters and repetition is not allowed, we use the standard permutation formula.
By definition, nPr = n!/(n-r)!
Substituting the values, we get:
9P4 = 9!/(9-5)!
= 9!/4!
= 362,880/120
9P4 = 3,024
This implies that there are 3,024 different possible finishing orders for the four positions.
Want to find the factorials quickly using an online tool? Use the factorial calculator.
Example 2: Permutation With Repetition
A school uses four-symbol locker codes. Each position can contain any of 6 available symbols, and a symbol may be used more than once. How many different locker codes are possible?
Solution
Step 1: Identify the total number of available items (n).
There are 6 symbols available for each position. Therefore, n = 6
Step 2: Identify the number of items being arranged (r).
Each locker code contains 4 symbols. Therefore, r = 4
Step 3: Determine whether repetition is allowed.
From the question, we know that the same symbol can appear more than once. This means repetition is allowed. Also, the order matters because a code such as 1234 is different from 4321.
Step 4: Apply the appropriate permutation formula.
When repetition is allowed, P = nr
Substituting n = 6 and r = 4 gives:
P = 64
= 1296
Therefore, there are 1,296 different locker codes.
When Should You Use Permutations?
Use permutations when you are selecting or arranging items and changing their order creates a different outcome.
Common examples include:
- determining first, second, and third place in a competition;
- assigning different positions to people;
- arranging books on a shelf;
- creating codes or passwords;
- arranging letters or numbers;
- scheduling people into ordered time slots; or
- determining possible finishing orders in a race.
Still struggling to determine whether to use a combination or permutation? Simply ask yourself, “Would changing the order of the selected items create a different outcome?” If the answer is yes, use a permutation. Otherwise, a combination is more appropriate.
Frequently Asked Questions
A permutation calculator is an online tool that finds the number of ways to arrange r items selected from n available items when order matters. Enter n and r to calculate nPr and see a step-by-step solution. You can also turn on the repetition option when the same item may be used more than once.
Without repetition, the permutation formula is P(n,r)= n!/(n-r)! where n is the total number of available items and r is the number being arranged. However, if repetition is allowed, use the formula P = n^r.
Ask whether changing the order of the selected items would create a different outcome. If the answer is yes, use a permutation. Otherwise, use a combination.
A permutation is used when order matters, while a combination is used when order does not matter. For example, assigning first, second, and third place is a permutation. Selecting three people for the same committee is a combination.
