Combination Calculator: nCr with and without Repetition
Count selections where order does not matter with nCr = n! / (r!(n − r)!) and with repetition, exactly for n up to 1000, with the formula shown, free in your browser.
Combinations (nCr)
Order does not matter, each item used at most once.
120
10C3 = 10! / (3! × (10 − 3)!)
Permutations (nPr)
Order matters, each item used at most once.
720
10P3 = 10! / (10 − 3)! = 10 × 9 × 8
Permutations with repetition (nʳ)
Order matters, items may repeat.
1,000
nʳ = 10^3
Combinations with repetition
Order does not matter, items may repeat.
220
C(n + r − 1, r) = C(12, 3)
Combination Calculator
This page opens with combinations first. Enter n, the number of items to choose from, and r, how many you pick, and the calculator gives nCr, the number of ways to choose when order does not matter, as an exact whole number. Below it you get combinations with repetition and the matching permutation counts, each with its formula. The page starts with n = 10 and r = 3, which gives 120.
The combination formula
nCr is read "n choose r". It counts the ordered arrangements nPr and then divides by r!, because each group of r items appears r! times in a different order.
How to use it
Steps
- Type n, the number of items, from 0 to 1000.
- Type r, the number chosen, from 0 to n.
- Read nCr at the top, and the other counts below it.
Worked examples
Lottery
Picking 6 numbers from 49: 49C6 = 13,983,816, so one ticket has a 1 in 13,983,816 chance of matching all six.
Poker hands
A 5-card hand from 52 cards: 52C5 = 2,598,960.
Ice cream, with repetition
3 scoops from 10 flavours, where the same flavour may be picked twice: C(10 + 3 − 1, 3) = C(12, 3) = 220.
Useful properties
Choosing r items to keep is the same as choosing n − r to leave, so nCr = nC(n − r): 10C3 = 10C7 = 120. Choosing none or all has one way, so nC0 = nCn = 1. The values of nCr for one n form a row of Pascal's triangle, and they are the coefficients of the binomial expansion (a + b)ⁿ. Results are exact big integers for n up to 1000; 1000C500 has 300 digits.
Combinations in probability
Combinations are the backbone of many probability questions. The chance of a particular outcome is the number of favourable selections divided by the number of possible ones. For example, the chance that a 5-card hand holds all four aces is 48 / 2,598,960, because the fifth card can be any of the other 48. Counting both numbers with nCr keeps the arithmetic exact.
Combinations versus permutations
If the same three people can be chosen in any order and it makes no difference, use nCr. If they take three different roles, or the order of the draw is recorded, use nPr. For 10 people and 3 places, the difference is a factor of 3! = 6: 120 groups but 720 ordered line-ups.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
The Pascal's Triangle Calculator shows nCr values as a triangle, and the full Permutation and Combination Calculator puts permutations first.
Frequently asked questions
What is a combination?
A selection where order does not matter. Picking 3 toppings from 10 is a combination: there are 120 ways, because each set of 3 can be ordered in 3! = 6 ways and 720 / 6 = 120.
What is the combination formula?
nCr = n! / (r!(n − r)!), also written as n choose r. For 52 cards and a 5-card hand, 52C5 = 2,598,960.
What is a combination with repetition?
When an item can be chosen more than once, such as scoops of ice cream from 10 flavours, the count is C(n + r − 1, r). For 3 scoops from 10 flavours that is C(12, 3) = 220.
Is nCr the same as a binomial coefficient?
Yes. nCr is the binomial coefficient C(n, r), the number in row n and position r of Pascal's triangle, and the coefficient of xʳ in (1 + x)ⁿ.
Why is nCr equal to nC(n − r)?
Choosing r items to take is the same as choosing n − r items to leave, so 10C3 = 10C7 = 120.