Statistics How To

Permutation Calculator / Combination 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:



13 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. Andale

    Hi, Frank,
    I’m not sure I understand your question completely but it might be the fundamental counting principle could help. 36 events (months), 3 ways to order each month so that’s 3^36.

    If you can clarify a bit and post on our forum, one of our mods will be happy to help!
    Stephanie

  3. 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.

  4. 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

  5. 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.

  6. Art

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

    Art

  7. 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!

  8. 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}

  9. 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.

  10. Andale

    Hello, Matthew,
    Unfortunately I’m not not able to answer all stats question here on the site. Could you please post your question on our forums?

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