Combination Calculator (nCr)
Count combinations C(n, r) exactly, with or without repetition, and see the matching permutations, the odds of one pick and a Pascal row.
A combination counts the different groups of r items you can pick from n when order does not matter: poker hands, lottery tickets, committees, pizza toppings. Enter n and r to get C(n, r) exactly, even when it runs to hundreds of digits, together with the matching permutation count, the odds of one particular pick and a full row of Pascal’s triangle for n up to 60.
How to use this tool
- Enter n, the number of items to choose from.
- Enter r, how many you pick.
- Tick “repetition allowed” if the same item can be chosen more than once, like scoops of ice cream.
- Read C(n, r), the permutation count and the odds.
Formula
C(n, r) = n! / (r! × (n − r)!) · with repetition: C(n + r − 1, r)
n! (n factorial) is n × (n − 1) × … × 1, and 0! = 1. Dividing by r! removes the r! orderings of each group, which is why combinations never exceed permutations P(n, r) = n! / (n − r)!. With repetition, the “stars and bars” argument turns the problem into choosing r positions from n + r − 1.
Example
A 5-card poker hand comes from a 52-card deck: C(52, 5) = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1) = 311,875,200 / 120 = 2,598,960 hands. If order mattered there would be 311,875,200 ordered deals, and the chance of being dealt one particular hand is 1 in 2,598,960.
Frequently asked questions
What is the difference between a combination and a permutation?
Order. Choosing Ana, Ben and Cal for a committee is one combination, but naming them president, secretary and treasurer gives 3! = 6 permutations. Use permutations when positions or order matter.
What are the odds of winning a 6/49 lottery?
There are C(49, 6) = 13,983,816 possible tickets, so one ticket has a 1 in 13,983,816 chance of matching all six numbers.
Why is C(n, r) the same as C(n, n − r)?
Choosing the r items to keep is the same as choosing the n − r items to leave out, so both counts match. For example, C(10, 3) = C(10, 7) = 120.
What does repetition allowed mean?
Items can be chosen more than once and order still does not matter. Picking 3 scoops from 5 flavours with repeats allowed gives C(7, 3) = 35 options.
Last updated: 2026-10-30