By · Updated

Permutation and Combination Calculator

nPr and nCr, With and Without Repetition

No ratings yet
Result

—

OneTask is free, with no sign-up, and built by one person.
If this tool saved you time, a coffee helps keep new ones coming.

How the Permutation and Combination Calculator works

This permutation and combination calculator counts the ways to choose r items from n. Combinations (nCr) ignore order, as in a lottery draw; permutations (nPr) count each order separately, as in first, second and third place. It also gives both counts with repetition allowed, and works exactly for n and r up to 10,000.

The one question to ask is: does order matter? Picking 3 pizza toppings from 10 is a combination, because pepperoni-ham-olive is the same pizza as olive-ham-pepperoni. Awarding gold, silver and bronze to 3 of 10 runners is a permutation, because the order changes the result. Repetition is the second question: a PIN can reuse digits, but a team cannot pick the same person twice.

How the permutation and combination calculator works

The formula for permutations is nPr = n! / (n − r)!, and for combinations nCr = n! / (r! × (n − r)!). Each combination can be ordered in r! ways, which links the two: nPr = nCr × r!. With repetition, permutations are n^r and combinations are (n + r − 1)Cr. Big integers keep every digit.

First example. 49C6 = 13,983,816, the number of possible 6-number lottery tickets from 49 balls.

Second example. 10P3 = 10 × 9 × 8 = 720 ways to award gold, silver and bronze among 10 runners.

How many 4-digit PINs are there? Digits can repeat and order matters, so 10^4 = 10,000.

Combinations and permutations for n from 6 to 30

n nC2 nC3 nC5 nC6 nP3
6 15 20 6 1 120
7 21 35 21 7 210
8 28 56 56 28 336
9 36 84 126 84 504
10 45 120 252 210 720
11 55 165 462 462 990
12 66 220 792 924 1,320
13 78 286 1,287 1,716 1,716
14 91 364 2,002 3,003 2,184
15 105 455 3,003 5,005 2,730
16 120 560 4,368 8,008 3,360
17 136 680 6,188 12,376 4,080
18 153 816 8,568 18,564 4,896
19 171 969 11,628 27,132 5,814
20 190 1,140 15,504 38,760 6,840
21 210 1,330 20,349 54,264 7,980
22 231 1,540 26,334 74,613 9,240
23 253 1,771 33,649 100,947 10,626
24 276 2,024 42,504 134,596 12,144
25 300 2,300 53,130 177,100 13,800
26 325 2,600 65,780 230,230 15,600
27 351 2,925 80,730 296,010 17,550
28 378 3,276 98,280 376,740 19,656
29 406 3,654 118,755 475,020 21,924
30 435 4,060 142,506 593,775 24,360

Tips and pitfalls

FAQ

What is the difference between a permutation and a combination?

A permutation counts arrangements where order matters; a combination counts selections where it does not. Choosing 2 of A, B, C gives 3 combinations (AB, AC, BC) but 6 permutations (AB, BA, AC, CA, BC, CB). Permutations are always at least as many as combinations.

How do you calculate combinations?

Use nCr = n! ÷ (r! × (n − r)!). For 5C2, that is 120 ÷ (2 × 6) = 10. In practice it is quicker to multiply r falling terms and divide by r!: 5 × 4 ÷ (2 × 1) = 10. The calculator does this exactly even when the answer has thousands of digits.

How many combinations of 6 numbers from 49 are there?

There are 13,983,816 ways to choose 6 numbers from 49, so a single ticket has a 1 in 13,983,816 chance of matching all six. If the order of the draw mattered, there would be 10,068,347,520 permutations, which is 720 times as many.

What does with repetition mean?

It means an item can be chosen more than once. A 3-letter code from 26 letters with repetition has 26^3 = 17,576 permutations. Choosing 3 scoops from 5 flavours, where two scoops can be the same, has 7C3 = 35 combinations with repetition.

How many ways can you arrange a deck of cards?

52! ≈ 8.07 × 10^67, a number so large that every properly shuffled deck is almost certainly in an order that has never existed before. Enter n = 52 and r = 52 to see it, or use the factorial calculator for the full 68 digits.

See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.

Related tools

Factorial Calculator

Exact n factorial up to 5000!, with digits and trailing zeros.

Probability Calculator

Combine two probabilities, and the odds of at least once.

Loot Drop Calculator

Your real odds of a rare drop, and how many tries you need.

Dice Roller

Roll D6, D20 or any custom die with a fair random generator.

Binomial Probability Calculator

Exactly, at most and at least k successes in n trials.