Combinations & Permutations Calculator

Written byOleh Soloninupdated 5 October 2026

n (total items)

r (chosen)

Result

Combinations C(n,r)120
Permutations P(n,r)720

Choosing versus arranging

Count how many ways you can choose or arrange r items from n. The difference is whether order matters — and it changes the answer hugely.

One question decides which formula you need: does the order matter? Picking three lottery numbers is a combination. Awarding gold, silver and bronze is a permutation.

The gap is dramatic. Choosing 3 people from 10 for a committee gives 120 possibilities; assigning those same 3 to distinct roles gives 720, because each group can be arranged six ways.

How to use

Enter the total number of items, n.
Enter how many are chosen, r.
Read the number of combinations and of permutations — the first ignores order, the second counts it.

Does the order matter

Choosing three from ten gives 120 combinations and 720 permutations. The ratio is exactly 3! = 6, because every combination can be arranged in six orders. That is the whole relationship between the two formulas.

Deciding which to use is a matter of whether order carries meaning. The membership of a three-person team is a combination; awarding gold, silver and bronze to those same three is a permutation.

The ratio is r!. Worked out with this calculator.
n and r	Combinations	Permutations	Ratio
10 choose 2	45	90	2
10 choose 3	120	720	6
10 choose 4	210	5,040	24

What this calculator shows

Combinations C(n,r), where order is irrelevant
Permutations P(n,r), where order matters
Exact whole-number results for n up to 1000: in full up to forty digits, beyond that the leading digits and how many there are

What to keep in mind

Items are selected without replacement — nothing is chosen twice.
Uses multiplicative formulas rather than raw factorials, so large n does not blow up.

FAQs

How do I tell which one I need?

Ask whether swapping two picks creates a different outcome. If it does, use permutations. A committee of {Ann, Bob} is the same as {Bob, Ann}; a president-and-deputy pairing is not.

What are the odds in a 6-from-49 lottery?

C(49,6) = 13,983,816 possible tickets, so one in roughly fourteen million.

Why does calculating factorials directly fail so fast?

A double-precision float holds factorials exactly only up to 22!, and 171! does not fit at all. Cancelling terms as you go keeps the intermediate values small; this calculator also works in exact integer arithmetic.

What if r is bigger than n?

Both counts are zero. You cannot choose five items from a set of three without repeating.

How many combinations of 10 choose 3?

120, against 720 permutations of the same numbers — six times more, because three items can be ordered in 3! = 6 ways.

Which one do I need?

Combinations when only the membership matters: a team, a hand of cards, a set of dishes. Permutations when order carries meaning: podium places, a passcode of distinct digits, a running order.

Worked example

A committee versus a podium

Input: n = 10, r = 3

Output: C(10,3) = 120 · P(10,3) = 720

Note: Exactly six times as many permutations, because each set of three can be ordered in 3! = 6 ways.

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Result

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