How do I count a lottery's possible draws?

Use combinations, because the order of the drawn numbers does not matter.

What does repetition allowed mean?

The same item can be chosen more than once, as in a PIN code.

Permutation & Combination Calculator

Count ordered and unordered selections, with and without repetition, using exact big-integer results.

Math Calculators
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Showing the example values. Change the inputs and press Calculate for your own result.

Combinations C(10, 3): order does not matter

120

Permutations P(10, 3): order matters = 720

Permutations nPr = P(10, 3): 720
Combinations nCr = C(10, 3): 120
Permutations with repetition (nʳ): 1,000
Combinations with repetition: 220
10! (all arrangements of n items): 3,628,800

Working

Permutations count ordered selections, so ABC and CBA are different.
Combinations count unordered selections, so ABC and CBA are the same.
C(n, r) = P(n, r) ÷ r!

About the permutation & combination calculator

Permutations count the ways to choose and arrange items when order matters; combinations count the ways to choose when order does not matter. This calculator gives nPr and nCr for any n and r, plus the counts when repetition is allowed and n factorial.

Results are exact, even when they have dozens of digits.

Who should use the permutation & combination calculator

Students in probability and discrete maths.
Anyone counting possible teams, codes or arrangements.
Game and puzzle designers.

How to use the permutation & combination calculator: what each input means

Total items (n): How many items there are to choose from, up to 1,000.
Items chosen (r): How many you pick.

How the permutation & combination calculator works: method and formula

Permutations without repetition: nPr = n! ÷ (n − r)!. Combinations: nCr = n! ÷ (r!(n − r)!). With repetition allowed, permutations become nʳ and combinations become C(n + r − 1, r). All values are calculated with exact big-integer arithmetic.

Permutations: P(n, r) = n! ÷ (n − r)!
Combinations: C(n, r) = n! ÷ (r! (n − r)!)

How to read your permutation & combination calculator result

If swapping two chosen items would count as a different result (a podium of first, second and third), use permutations. If not (a committee), use combinations. Combinations are always the permutations divided by r!.