Permutation & Combination Calculator

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Calculate permutations (nPr) and combinations (nCr) for n items taken r at a time, with the underlying factorials. Exact big-integer math. Free and instant.

The size of the whole set.
How many are selected (r ≤ n).
Combinations (nCr) –
Permutations (nPr) –
n! –
r! –
(n − r)! –
Permutations count ordered selections (nPr = n! ÷ (n−r)!); combinations count unordered ones (nCr = n! ÷ (r!(n−r)!)). n and r must be whole numbers with 0 ≤ r ≤ n. Very large results are shown in scientific notation.

About Permutation & Combination Calculator

This permutation and combination calculator computes nPr and nCr for a set of n items taken r at a time, along with the underlying factorials n!, r! and (n − r)!. Permutations count ordered selections; combinations count selections where order does not matter.

The formulas are the classic ones: nPr = n! ÷ (n − r)! and nCr = n! ÷ (r! × (n − r)!). The tool uses exact big-integer math, so results are precise even for large n rather than rounded floating-point approximations; very large values are also shown in scientific notation.

It is handy for probability homework, lottery odds, counting passwords or seating arrangements, and any combinatorics problem where you need to know how many ways r items can be chosen from n.

How to use Permutation & Combination Calculator

  1. Enter the total number of items n — the size of the whole set.
  2. Enter r, how many items are selected. It must satisfy 0 ≤ r ≤ n.
  3. Read the combinations (nCr) and permutations (nPr) instantly.
  4. Check the factorial breakdown — n!, r! and (n − r)! — to see how the results are built.

Frequently asked questions

Permutations count arrangements where order matters — ABC and CBA are different. Combinations ignore order, so ABC and CBA count once. That is why nPr is always at least as large as nCr.

You cannot choose more items than the set contains. Both formulas require whole numbers with 0 ≤ r ≤ n; the tool flags any input where r exceeds n and asks you to lower it.

Yes. The calculator uses arbitrary-precision integer arithmetic, so factorials with hundreds of digits are computed exactly. When a value gets very long, it is additionally displayed in scientific notation for readability.

Both equal 1. There is exactly one way to choose nothing from a set, and 0! is defined as 1, which keeps the formulas consistent for every valid r.

Anywhere order is irrelevant: lottery number picks, poker hands, choosing a committee from a team, or counting pizza-topping mixes. Use nPr instead for rankings, passwords and seating orders where sequence matters.
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