How it works
A combination is a selection where order does not matter. A 5-card poker hand is a combination: the same five cards are the same hand no matter the order they were dealt. There are 2,598,960 possible 5-card hands from a 52-card deck.
Without repetition, each item can be chosen at most once, giving n! ÷ (r!(n − r)!). With repetition allowed, such as choosing 3 scoops from 10 flavors where you can repeat a flavor, the count is C(n + r − 1, r).
The reciprocal of the count is the chance of one specific combination if every combination is equally likely, which is how lottery odds are calculated.
Formula
nCr = n! ÷ (r! × (n − r)!) · With repetition: C(n + r − 1, r)
Questions people ask
How are lottery odds calculated?
For a draw of 6 numbers from 49, the number of tickets is 49C6 = 13,983,816, so the jackpot odds are 1 in 13,983,816. Games with a bonus ball multiply by the bonus pool size.
Why is nCr equal to nC(n − r)?
Choosing r items to include is the same as choosing n − r items to leave out, so both counts are identical. For example, 10C3 = 10C7 = 120.
How do combinations relate to permutations?
nCr = nPr ÷ r!, because each group of r items can be ordered in r! ways and combinations count all of those orders once.