Statistics How To

Permutation Calculator / Combination Calculator

Calculators > Permutation Calculator / Combination Calculator

Watch the video or scroll down for the calculator.

The “Mathy” Way



When the order of items matters, that’s called a Permutation.

When the order of items doesn’t matter, that’s called a Combination.
Since we are not allowed to repeat items, we use the following formula:

Number of possible Permutations  
=    nr
 
=   
 
Number of possible Permutations  
=    n!


(n – r)!
 
=    !


()!
 
Number of possible Combinations  
=    (n + r – 1)!


r!(n – 1)!
 
=    ( + – 1)!


!( – 1)!
 
Number of possible Combinations  
=    n!


r!(n – r)!
 
=    !


!()!
 

The Visual Way

A form of the permutation problem that students commonly see is the “committee” problem. For example:

If there are 5 people, Jim, Jane, Bob, Susan, and Ralph, and only 3 of them can be on the new PTA committee, how many different combinations are possible?

In this example, there are 5 people to choose from (so n equals 5), and we need to choose 3 of them (so r equals 3).

Order doesn’t matter: if Jim is on the committee, he’s on the committee whether he’s picked first or last. Repetition isn’t allowed because Susan can’t be on the committee twice (even if she really wants to be!)

So, if we use the “mathy” way from above, we know the formula is:

Number of possible Combinations  
=    n!


r!(n – r)!

And we input the number 5 for n, and 3 for r, and so we know that there 10 possible combinations. But what does that actually mean?

Combinations Generator

I’ll show you using our generator:

All possible items:

  • No.
  • No.

Possibilities:



If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Permutation Calculator / Combination Calculator was last modified: October 15th, 2017 by Stephanie

78 thoughts on “Permutation Calculator / Combination Calculator

  1. Frank

    I am interested in how many choices a person would have to divide 36 months among 3 people?
    Thanks!

  2. Frank

    Here is the problem.
    A person has 36 months to divide among 3 people.
    How many different combinations could there be.
    For example one combo would be 12 months each.

  3. Art

    I like the calculator, however, my problem would need a slightly different formula that I haven’t found.
    PROBLEM: I have 25 choices, and want to know the total number of combinations possible. Since I want combinations, not permutations, order does not matter. To keep it simple, repetition is not allowed. The combinations can be of any number of choices (r=1,2,3,…25). I can use the calculator and solve sequentially for r=1, r=2, r=3, etc., then sum the answers. However, there should be an equation that would solve the problem in one calculation. Can you show me that equation?

    Thank you,

    Art

  4. Andale

    Well, Art. The calculator uses the combinations formula (n!/(n-k!)k!). As you want to know k for all possible choices (1 to 25). I don’t know of a single equation other than a summation equation (Σ (n!/(n-k!)k! from n=1 to n=25)) I would suggest Excel. Copy the formula to 25 cells and then use autosum.

  5. Art

    OK, thanks. I appreciate the quick reply, and will probably use the spreadsheet method you suggested.

    Art

  6. Rachel G.

    So if i have a 4-input combination lock that requires pink, green or purple as the input options. How many combinations are possible and can you list them? The order doesnt matter and they can be repeated. It would be really amazing if i could get this lock open! Ive been trying with no luck all weekend. Thanks in advance!

  7. Andale

    81

    {pink,pink,pink,pink} {pink,pink,pink,green} {pink,pink,pink,purple} {pink,pink,green,pink} {pink,pink,green,green} {pink,pink,green,purple} {pink,pink,purple,pink} {pink,pink,purple,green} {pink,pink,purple,purple} {pink,green,pink,pink} {pink,green,pink,green} {pink,green,pink,purple} {pink,green,green,pink} {pink,green,green,green} {pink,green,green,purple} {pink,green,purple,pink} {pink,green,purple,green} {pink,green,purple,purple} {pink,purple,pink,pink} {pink,purple,pink,green} {pink,purple,pink,purple} {pink,purple,green,pink} {pink,purple,green,green} {pink,purple,green,purple} {pink,purple,purple,pink} {pink,purple,purple,green} {pink,purple,purple,purple} {green,pink,pink,pink} {green,pink,pink,green} {green,pink,pink,purple} {green,pink,green,pink} {green,pink,green,green} {green,pink,green,purple} {green,pink,purple,pink} {green,pink,purple,green} {green,pink,purple,purple} {green,green,pink,pink} {green,green,pink,green} {green,green,pink,purple} {green,green,green,pink} {green,green,green,green} {green,green,green,purple} {green,green,purple,pink} {green,green,purple,green} {green,green,purple,purple} {green,purple,pink,pink} {green,purple,pink,green} {green,purple,pink,purple} {green,purple,green,pink} {green,purple,green,green} {green,purple,green,purple} {green,purple,purple,pink} {green,purple,purple,green} {green,purple,purple,purple} {purple,pink,pink,pink} {purple,pink,pink,green} {purple,pink,pink,purple} {purple,pink,green,pink} {purple,pink,green,green} {purple,pink,green,purple} {purple,pink,purple,pink} {purple,pink,purple,green} {purple,pink,purple,purple} {purple,green,pink,pink} {purple,green,pink,green} {purple,green,pink,purple} {purple,green,green,pink} {purple,green,green,green} {purple,green,green,purple} {purple,green,purple,pink} {purple,green,purple,green} {purple,green,purple,purple} {purple,purple,pink,pink} {purple,purple,pink,green} {purple,purple,pink,purple} {purple,purple,green,pink} {purple,purple,green,green} {purple,purple,green,purple} {purple,purple,purple,pink} {purple,purple,purple,green} {purple,purple,purple,purple}

  8. matthew

    Ok so i have a lock that i need the combo to, i had it in third grade and the numbers are only 1-9 with the usual, only three in the combination. I need all the possible combos w/ repeating numbers hoping i can see the combo and remember and if not go through all the numbers manually.

  9. Al

    Hello! I really like that the combinations generator physically shows all of the combinations. However, I need a slightly different calculator. I need to calculate and show all the combinations that 15 people can be in two groups (one of 7 and one of 8).

  10. Jefferson

    This calculator works great because its assuming each N has the EXACT same number of R. What if some N have R=4 and some N have R=5?

  11. Andale

    In that case the combinations formula itself would chance, so you’d have to do two computations (the calculator is based on the combinations formula).

  12. Andy

    Thank you for your site. Could you tell me the formular for calculating possable combinations for 2 numbers and 2 letters Eg starting with 00AA

    Thank you in advance for your help

    Andy

  13. Robert williams

    I have a locking device with 10 buttons which can all be pressed up to 3 times. The buttons can be pressed in any order, therefore the locking device is a combination lock. The user can set the code using any number of buttons from just one button to all the buttons.
    I am trying to establish how many combinations they would be for this type of device.

    Any help would be welcome.

    Rob.

  14. EUGENE JACK

    i think that it;s fabulous that you came up with such an impressive way of calculating things,
    loved it

  15. Eric McCrea

    I have a lock that consists of 5 numbers. Please help. The numbers are 56890. I need every different order to try and unlock it. Please help and thank you in advance.

  16. Jacob

    Can you please help me to find the right way to calculate this problem I have. I have 12 people in a group and in need to find the total number of possible pairs of 2. If what I have done is correct would it be 66?

    Thanks in advance

  17. sherri

    hi
    i would like to find out how many different 12 digit combinations can you get using only numbers 1,2,3 and could you please list these different combinations for me.

  18. Dwain

    Is it possible for me to get a list of all five set combination from 1 and 22
    eg 3,8,22,12,9
    3,10,21,8,9

  19. Oljeg

    Hi, I am trying to find the solution (formula) for doing the following:
    For any number of players (divisible by three), I need to calculate a number of rounds (in my case 8) for three players to play each other without any repetition.
    So the first round of etc. 21 players would be:
    1,2,3 4,5,6, 7,8,9 10,11,12 13,14,15 16,17,18 19,20, 21 and I am looking for the combinations for the next 7 rounds so that no one plays the same player again.

    Regards,
    Oljeg

  20. falco

    hi,
    I am amazed with your calculator, exactly what i needed, if i want to do the same calculator to excel, can you help me with the formula?

  21. Brian Larson

    How would I calculate the different sets of 9 numbers with 3 numbers to a group, where no 2 numbers can be in the same group of 3 more than once?
    An example set of 12 which works is 123, 456, 789, 147, 258, 369, 159, 267, 348, 168, 249, 357.

  22. Andale

    If I’m understanding your problem correctly, then you would just calculate combinations without repetition.

  23. jim neville

    lhow many combinations of 6 numbers are there in 19 numbers – each number to feature only once in each combination?

  24. Victor

    Combinations & Permutation Calculator gives me the result of n=8, r =4 or 330 combinations. Is there an easy way to take those 330 combinations and separated them into groups by sum. For example show only combinations that add up 14, 15, 21, etc.

  25. Andale

    I would put them into an Excel spreadsheet and use the sort function to sort in ascending order. Then you can highlight by group…

  26. Pius

    Hello.. ..pls I want you to help me permutate 10 football teams to 2 event outcomes (that’s odd and even) goals

  27. Andale

    Hi, Pius,
    I’m not sure exactly what you are asking. It’s possible to have (theoretically) an infinite amount of goals (1…infinity and 2…infinity) no matter how many teams you have. Could you please rephrase your question?

  28. Dave

    Hi,

    I have a combination lock that I need all options for.

    23-59-79 numbers in that order but each or all could be + or – 1 number. Can you help?

    Thank you
    Dave

  29. Andale

    I don’t know of a formula. I would write them out as there aren’t many, e.g 22 59 79, 23 59 79, 24 59 79…

  30. Ryan

    Hey I would like to know how many combinations there would be for a straight trifecta using 9 runners from the race, they are: Hartnell, Jameka, Oceanographer, HeartbreakCity, Almandin, BondiBeach, BigOrange, WicklowBrave, SecretNumber
    Cheers

  31. Larry Burnette

    I went in the hospital for surgery,went,wentinto a coma, and forgot my combination to my safe the three numbers i have 07-09-42# just need how many combinations and what they are for these three numbers. HELP

  32. Andale

    Do you mean combinations of the letters in the words? Choosing all of them at the same time? Assuming the items do not repeat and that the order matters, that’s 362880 possibilities.

  33. Kayemdee

    Hi.
    I have seven people to be on an out of hours duty roster. It needs to change every week on a repeating cycle. I need two people at any one time out of the pool of seven to be on call per week. How do I work out how they all do an equal amount of call duty and have as much time off as possible till they are back on call.

  34. Andale

    :
    Scroll down to Combinations Generator. If you input the names, it will give you the list of possibilities. Then just group the results manually to address max time between calls.

  35. Knight

    I have 17 numbers and want to select 5 numbers from them, how to I get the all the combination to this please and permutations please many thanks

    1 4 8 10 13 17 19 22 26 28 31 35 37 40 44 46 49

  36. JacqK

    You want to form a 5 letter code word using the letters A, B, C, D, E and numbers
    1, 2, 3, 4, 5 and allowing for repetition of letters and numbers.
    (a) How many different code words can you form?
    (b) What is the probability that a letter or a number is repeated?
    (c) What is the probability that no number is repeated?

  37. Andale

    Hi, Jacq,
    You can use the calculator to answer (a).
    For b and c, use the combinations generator (scroll down). Find your outputs and from there you should be able to figure out the probabilities.

  38. Rose

    I am looking for all of the possible 5 and also 6 letter words that can come from the letters, C, C, H, O, R, U, S.
    They may be in ANY order. This would have all words beginning with C through S. The calculator only shows words beginning with the 1st C. Since there are 2 C’s, then a word may contain 2 C’s, but not all words will have 2 C’s.
    How can I find the correct word combinations?

  39. Andale

    Rose, I typed in C,C,H,O,R,U,S into the generator at the bottom and it did come up with a list that included two CCs. That said, it isn’t going to give you words — the letters will still be scrambled.
    C, C, H, O, R
    C, C, H, O, U
    C, C, H, O, S
    C, C, H, R, U
    C, C, H, R, S
    C, C, H, U, S
    C, C, O, R, U
    C, C, O, R, S
    C, C, O, U, S
    C, C, R, U, S
    C, H, O, R, U
    C, H, O, R, S
    C, H, O, U, S
    C, H, R, U, S
    C, O, R, U, S
    C, H, O, R, U
    C, H, O, R, S
    C, H, O, U, S
    C, H, R, U, S
    C, O, R, U, S
    H, O, R, U, S

  40. Rose

    That is okay, since you can use any of the set letters to make a word. The letters may be in any order.
    When using the letters C,C,H,O,R,U,S, they may be as such:
    CHORUS
    OCCURS
    There is nothing saying the letters must be in the same order.
    Thank you for your help.

  41. Therese

    Hi,

    I’ve tried playing around with the calculator to do this but couldn’t work out how to get the two subsets in the answer.

    I have nine students that I need to divide into two groups (one of four and one of five). I’d like to see all the possible grouping combinations.

    The names are JackT, JackA, Tess, Carla, Will, Sienna, Lorenzo, Tom, Sam

    Thank you so much for your help!
    Therese :o)

  42. Andale

    What if you choose four from the nine using the generator)? The remainder will give you the five group. For example (JackT, JackA, Tess, Carla), leaves Will, Sienna, Lorenzo, Tom, Sam.

  43. Henry

    How many possible combinations are there in a bet involving 13 football games with each game having a possible outcome of win, draw, or lose?

  44. Beck

    Trying to work this out…

    I want to add 6 numbers together to equal 45 or less – I’d like to know how many combinations if the 6 numbers I’m adding together are between 1 and 15. I guess order doesn’t matter as long as the outcome is 45 or less. The last thing is no more than 2 of the 6 numbers can be repeated repeated in the one equation.

  45. Jon

    I have combo lock need 4 digit number anything from 0-9 numbers can repeat no specific order and also 4 digit combo with dates 1370 4013 2363 8013

  46. Loran

    I have 1a ,1b , 2a ,2b , 3a ,3b , 4a ,4b , 5a ,5b , number cannot repeat , letters yes , order doesn’t matter .
    How should I do it?
    Thanks

  47. Andale

    Loran,
    What’s your question? Are you asking how to put this into the calculator? Or is something not working right?

  48. Andale

    Put them into the generator at the bottom of the page. You’ll get 120 possibilities.
    Then just delete the possibilities where there are repeated numbers. The first 9 are
    1a, 1b, 2a
    1a, 1b, 2b
    1a, 1b, 3a
    1a, 1b, 3b
    1a, 1b, 4a
    1a, 1b, 4b
    1a, 1b, 5a
    1a, 1b, 5b
    1a, 2a, 2b
    So can delete all of those!

  49. Loran

    Thanks but I need some kind of a formula as I want to use it on different combinations like n= 10 , r =6 …etc

  50. Loran

    I have numbers 1 to 5, each has 2 possible outcomes : a or b .
    R= 3

    so I need all combination where the numbers don’t repeat but letters can.

    Example : 1a , 2a ,3b
    1a, 2b, 3a……………………

  51. andy

    hi i’m trying to make rotating groups of three for a class where the goal is that no two people are ever in the same group. do you have any way to do that?

  52. Andale

    I would figure out all the possible combinations first. Then just delete groups where there are duplicates.

  53. andy

    i’ve been trying to do that but with a longer list 10+, i find the bottom of the list has only repeats. maybe the order of the list is not random enough. For the 13 people. i go down and get stuck.
    1. SY, CD, JN
    2. SY, HD, NMG
    3. SY, HK, BJ
    4. CD, HD, HK
    5. CD, NMG, BJ
    6. JN, HD, NMG
    7. JN, HK, SH
    8. BJ, SH, SJZ
    9. SH, HF, WH
    10. SJZ, HF, GZ
    11. HF, WH, GZ
    12. HF, WH, XA
    13. HF, GZ, XA
    14. WH, GZ, XA