How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (2024)

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1Calculating Permutations without Repetition

2Calculating Permutations with Repetition

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Last Updated: March 17, 2024References

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If you're working with combinatorics and probability, you may need to find the number of permutations possible for an ordered set of items. A permutation is an arrangement of objects in which the order is important[1] (unlike combinations, which are groups of items where order doesn't matter[2]). You can use a simple mathematical formula to find the number of different possible ways to order the items. To start off, you just need to know whether repetition is allowed in your problem or not, and then pick your method and formula accordingly.

Method 1

Method 1 of 2:

Calculating Permutations without Repetition

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  1. 1

    Start with an example problem where you'll need a number of permutations without repetition. This kind of problem refers to a situation where order matters, but repetition is not allowed; once one of the options has been used once, it can't be used again (so your options are reduced each time).[3]

    • For instance, you might be selecting 3 representatives for student government for 3 different positions from a set of 10 students. No student can be used in more than one position (no repetition), but the order still matters, since the student government positions are not interchangeable (a permutation where the first student is President is different from a permutation where they're Vice President).
    • This kind of problem is often labeled as How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (5) or How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (6), where How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (7) is the number of total options you have to choose from and How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (8) is how many items you need to choose.

  2. How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (9)

    2

    Know the formula: How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (10). In the formula, How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (11) is the number of total options you have to choose from and How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (12) is how many items you need to choose, where order matters and repetition is not allowed.

    • In this example, How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (13) would be the total number of students, so How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (14) would be 10, and How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (15) would be the number of people chosen, so How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (16) would be 3.

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  3. How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (17)

    3

    Plug your numbers in for How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (18) and How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (19).

    • In this case you'd have How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (20).

  4. How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (21)

    4

    Solve the equation to find the number of permutations.

    • If you have a calculator handy, find the factorial setting and use that to calculate the number of permutations. If you're using Google Calculator, click on the x! button each time after entering the necessary digits.
    • If you have to solve by hand, remember that, for each factorial, you start with the main number given and then multiply it by the next smallest number, and so on until you get down to 0.
    • For example, you would calculate 10! by doing (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1), which gives you 3,628,800 as a result. 7! would be (7 * 6 * 5 * 4 * 3 * 2 * 1), which would equal 5,040. You'd then calculate 3,628,800/5,040.
    • In the example, you should get 720. That number means that, if you're picking from 10 different students for 3 student government positions, where order matters and there is no repetition, there are 720 possibilities.

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Method 2

Method 2 of 2:

Calculating Permutations with Repetition

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  1. How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (23)

    1

    Start with an example problem where you'll need a number of permutations where repetition is allowed.

    • For example, if you have 10 digits to choose from for a combination lock with 6 numbers to enter, and you're allowed to repeat all the digits, you're looking to find the number of permutations with repetition.
    • A permutation with repetition of n chosen elements is also known as an "n-tuple".[4]

  2. How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (24)

    2

    Know the formula: How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (25). In this formula, n is the number of items you have to choose from, and r is how many items you need to choose, in a situation where repetition is allowed and order matters.[5][6]

    • In the example, How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (26) is How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (27), and How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (28) is How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (29).

  3. How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (30)

    3

    Plug in How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (31) and How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (32).

    • In the example, you'll get the equation How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (33).

  4. 4

    Solve for the number of permutations. If you have a calculator handy, this part is easy: Just hit 10 and then the exponent key (often marked xy or ^), and then hit 6.

    • In the example, your answer would be How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (35). This means that, if you have a lock that requires the person to enter 6 different digits from a choice of 10 digits, and repetition is okay but order matters, there are 1,000,000 possible permutations.

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    Is there an easy way in order to memorize the formula for this?

    How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (36)

    Alex

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    Yes. Think of it like this: subtract the total amount by the total items. For example 7 and 4. Then, find 7! to 4, so 7x6x5 and then find the answer, and you’ll get the permutations.

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      Tips

      • Some graphing calculators offer a button to help you solve permutations without repetition quickly. It usually looks like nPr. If your calculator has one, hit your How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (37) value first, then the permutation button, and then your How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (38) value.

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      About This Article

      How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (53)

      Reviewed by:

      Joseph Meyer

      Math Teacher

      This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 76,287 times.

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      Updated: March 17, 2024

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      How to Calculate Permutations: 8 Steps (with Pictures) - wikiHow (2024)

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