The rule
C(n, k) = n! ÷ (k! × (n − k)!)
Why it works
A combination counts selections where order is irrelevant. Start from the number of ordered selections and divide by k!, since every group of k items can be ordered in k! ways and all of those count as one.
How to do it by hand
- Work out how many ordered selections there are
- Divide by k!, the number of ways to order each group
- The result counts each group exactly once
- Choosing k is the same as leaving out n − k
What is worth knowing
The default here, 6 from 49, is the standard lottery: 13,983,816 combinations, so one ticket has that chance of winning. The symmetry C(n,k) = C(n,n−k) is worth knowing because it halves the work — choosing 47 from 49 is the same count as choosing 2, and the second is far easier to compute. Combinations with repetition are a different formula and a different question: how many ways to pick k things from n types when you can take several of the same, which is the 'stars and bars' problem.