Combinations and permutations both count ways to choose items from a set — the entire difference comes down to one question: does the order of selection matter?
Combinations: order doesn't matter
C(n, r) counts how many ways you can choose r items from a set of n when the order of selection is irrelevant — picking {A, B, C} is the same outcome as picking {C, B, A}. The formula is C(n, r) = n! ÷ (r! × (n − r)!).
Worked example: lottery odds
Choosing 6 numbers out of 49, where the order you pick them in doesn't change your ticket, gives C(49, 6) = 13,983,816 possible combinations — this is exactly why 6/49-style lottery odds are famously long. A smaller example: choosing 3 items from 10 gives C(10, 3) = 120 ways.
When order does matter: permutations
If arranging the same items in a different order counts as a different outcome (like assigning 1st, 2nd and 3rd place in a race), you need a permutation, not a combination — and the count is always larger, since every unique combination can usually be arranged in multiple orders.
Common mistakes
- Using the combination formula for a ranking or ordering problem — this undercounts, because it treats different orderings of the same items as one outcome.
- Assuming a larger n or r always means a dramatically larger result — the count grows fastest when r is close to n/2, and shrinks again as r approaches n.
Calculate any C(n, r) with the Combination Calculator.