Permutation & Combination Calculator

Permutations count arrangements where order matters, and combinations count selections where order does not. This calculator finds nPr and nCr exactly using whole-number arithmetic, so even very large results are precise. It also gives the counts when repetition is allowed and explains why combinations divide by r!.

Try:

How to use the permutation & combination calculator

  1. Enter the total number of items n.
  2. Enter how many are chosen, r.
  3. Press Calculate to see nPr, nCr and the steps.

Formula

nPr=n!(n−r)!{}^{n}P_{r} = \frac{n!}{(n-r)!}
(nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}

Worked example: n = 10, r = 3

  • Total items n: 10
  • Items chosen r: 3

✓ Answer checked

10P3=720,(103)=120{}^{10}P_{3} = 720,\quad \binom{10}{3} = 120
With repetition (order matters)
10001000
With repetition (order does not matter)
(123)=220\binom{12}{3} = 220
Step-by-step working (2 steps)

  1. Permutations (order matters) nPr = n!/(n − r)!

    Multiply 3 decreasing numbers starting from 10.

    nPr=n!(n−r)!=10×9×8=720{}^{n}P_{r} = \frac{n!}{(n-r)!} = 10 \times 9 \times 8 = 720

  2. Combinations (order does not matter) nCr = n!/(r!(n − r)!)

    Divide by r! = 6 because each group of 3 can be arranged in r! orders.

    (nr)=nPrr!=7206=120\binom{n}{r} = \frac{{}^{n}P_{r}}{r!} = \frac{720}{6} = 120
Formulas used
nPr=n!(n−r)!{}^{n}P_{r} = \frac{n!}{(n-r)!}
(nr)=n!r! (n−r)!\binom{n}{r} = \frac{n!}{r!\,(n-r)!}
How this was checked
  • ✓ The step-by-step multiplicative formula gives the same number of combinations.

Frequently asked questions

What is the difference between a permutation and a combination?

In a permutation the order matters (first, second, third place). In a combination only the group matters (a team of three).

How many ways are there to choose 6 numbers from 49?

13,983,816 combinations, which is 49C6.

Why divide by r! for combinations?

Each group of r items can be arranged in r! different orders, and a combination counts all of those as one.

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