Combination Calculator (nCr)
Enter how many items you have and how many you pick. Order does not matter, so ABC and CBA count as the same choice.
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Formula
nCr = n! ÷ (r! × (n - r)!) = nPr ÷ r!
Worked example
Choose 5 cards from 52: 52C5 = (52 × 51 × 50 × 49 × 48) ÷ 5! = 311,875,200 ÷ 120 = 2,598,960.
A larger case: 56C23 = 3,167,295,784,216,200. Results above 9,007,199,254,740,991 are shown as 7 significant digits plus the exact digit count, for example 170C85 = 9.144842 × 10^49 (50 digits).
Frequently asked questions
What is the combination formula?
nCr = n! ÷ (r! × (n - r)!). For 5C2 that is 120 ÷ (2 × 6) = 10.
How many lottery combinations are there for 6 numbers out of 49?
49C6 = 13,983,816, so one ticket has a 1 in 13,983,816 chance of matching all six numbers.
Why is nCr the same as nC(n - r)?
Choosing r items to take is the same as choosing n - r to leave behind. So 20C17 = 20C3 = 1,140.
Why is n limited to 170?
170! is about 7.26 × 10^306, the largest factorial a standard double-precision number can hold, so larger n is rejected. Answers are computed exactly with integer arithmetic. Up to 9,007,199,254,740,991 they are shown in full; larger ones are shown as 7 significant digits, such as 170C85 = 9.144842 × 10^49, together with the exact number of digits, so no digit is invented.