functions Combination & Permutation Calculator star

Combination vs. Permutation, and How They're Calculated

Both formulas count the number of ways to choose r items out of a group of n, but they answer a different question depending on whether the order of selection matters.

In a permutation, order matters — swapping two chosen items produces a different, distinct outcome. The formula is P(n,r) = n! ÷ (n−r)!. In a combination, order does not matter — the same group counts only once no matter how it was arranged. The formula is C(n,r) = n! ÷ (r! × (n−r)!), which is just the permutation formula divided by r! to cancel out the ways of reordering the same r items.

Worked example: from 5 people, choosing 3 to fill a podium — 1st, 2nd, 3rd place — is a permutation, because Alice-Bob-Carol finishing 1st-2nd-3rd is a different result from Bob-Alice-Carol. P(5,3) = 5!/(5-3)! = 120/2 = 60 possible orderings. Choosing the same 3 people to form an unordered 3-person committee is a combination — Alice, Bob and Carol as a committee is the same committee regardless of who was named first. C(5,3) = 5!/(3!×2!) = 120/12 = 10 possible committees. Both use the same 5 people and pick 3, but the different question — does the arrangement matter? — changes the answer from 60 to 10.

The tool also computes plain factorials (n!), the building block behind both formulas: n! is the product of every integer from 1 to n, and it grows extremely fast (10! is already 3,628,800). For n large enough that a regular calculator would run out of precision, this tool uses exact big-integer arithmetic instead of floating-point math, so results up to n = 5000 come out precise to the last digit rather than in rounded scientific notation.

What You Should Know

• The quickest test: ask "if I reorder the same items, is it a different outcome?" If yes, use permutation; if no, use combination. Lottery numbers, committees, hands of cards and ingredient lists are combinations. Passwords, race rankings, and podium finishes are permutations.
• 0! = 1 by definition, not by calculation — there is exactly one way to arrange zero items (do nothing), and this convention keeps formulas like C(n,n) = n!/(n!×0!) = 1 correct: there is only one way to choose an entire group.
• C(n,r) always equals C(n, n−r) — choosing which 3 of 10 people join a committee is the same count as choosing which 7 stay out, since both describe the same split.
• A frequent mistake is using the permutation formula for a genuinely unordered situation (like a raffle draw), which inflates the count by a factor of r! — always confirm whether swapping the order changes the outcome before picking a formula.
• Scale: this tool handles n up to 5000 with exact big-integer results, far past where ordinary floating-point calculators start rounding or overflowing.