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Combination Calculator
A combination is a selection where order does not matter. Enter the total number of items n and how many you are choosing r to get nCr, “n choose r”, exactly — plus the count when items can be chosen more than once and the matching number of permutations.
- Formula
- 52! ÷ (5! × (52 − 5)!) = 52! ÷ (5! × 47!)
- With repetition allowed
- 3,819,816
- Permutations (nPr), if order mattered
- 311,875,200
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Formula
How to use it
- Enter the total number of distinct items, n.
- Enter how many are chosen, r.
- Read nCr. Results over 30 digits are shown in scientific notation.
Worked examples
Five-card hands from a 52-card deck
- Combinations (nCr)
- 2,598,960
- Permutations (nPr), if order mattered
- 311,875,200
Choosing 6 numbers from 49
- Combinations (nCr)
- 13,983,816
A committee of 3 from 10 people
- Combinations (nCr)
- 120
- Formula
- 10! ÷ (3! × (10 − 3)!) = 10! ÷ (3! × 7!)
- With repetition allowed
- 220
Lottery and card odds
Choosing 6 numbers from 49 can be done 13,983,816 ways, so one ticket has a 1 in 13,983,816 chance of matching all six. Powerball’s five white balls from 69 give 11,238,513 combinations; multiplied by the 26 possible Powerballs that is 1 in 292,201,338 for the jackpot.
A standard deck has 2,598,960 different five-card hands.
Useful properties
Choosing r items to take is the same as choosing n − r to leave behind, so nCr = nC(n − r): 52C5 = 52C47.
With repetition allowed — three scoops from 10 ice-cream flavors, where a flavor can be picked twice — the count is C(10 + 3 − 1, 3) = 220 instead of 120.
Questions people ask
What is 52 choose 5?
2,598,960 — the number of five-card poker hands.
What is 10 choose 3?
120, from (10 × 9 × 8) ÷ (3 × 2 × 1).
What is the difference between a combination and a permutation?
A combination ignores order; a permutation counts each ordering separately. Three items chosen from ten make 120 combinations but 720 permutations.