Combinations Calculator
Enter the size of your set and how many you pick, and get the number of ways to choose them when order does not matter — the count behind lotteries, committees, and card hands.
Two values, one count
Enter the total number of items (n) and how many you choose (r); the calculator returns C(n, r) with the formula C(n, r) = n! ÷ (r! × (n − r)!).
Order is ignored
Combinations count selections, not arrangements — picking the same items in a different order is counted only once.
What is a combination?
Choosing a group when order does not matter
A combination is a way of choosing r items from a set of n items where the order of the choice does not matter — the group is the same combination however you list it. The number of such selections is written C(n, r), read "n choose r", and is given by the formula C(n, r) = n! ÷ (r! × (n − r)!). Combinations answer "how many different groups can I form", which is exactly the question behind lottery odds, committee selection, and the hands you can be dealt from a deck of cards.
Enter your total items and how many you choose to get the number of combinations instantly.
One formula, built from the set size n and the number chosen r.
C(n, r) = n! ÷ (r! × (n − r)!)The factorial n! multiplies every whole number from 1 up to n. Dividing n! by r! removes the orderings within your chosen group, and dividing by (n − r)! removes the orderings of the items left behind — what remains is the number of distinct groups. In practice this tool uses the equivalent multiplicative form so even large values of n never overflow through an intermediate n!.
Suppose you want to choose 3 students from a class of 10 to form a team.
Identify n and r
n = 10 total students, r = 3 chosen — order does not matter for a team.
Apply the formula
C(10, 3) = 10! ÷ (3! × 7!) = 3,628,800 ÷ (6 × 5,040).
Simplify
3,628,800 ÷ 30,240 = 120 — there are 120 possible teams.
The number you get is the count of distinct groups you can form, with every arrangement of the same items counted only once. The first thing to notice is that combinations are always fewer than permutations: because each group of r items can be ordered in r! different ways, P(n, r) = C(n, r) × r!, so order-sensitive counts are always larger. The result is also symmetric — C(n, r) = C(n, n − r) — because choosing which r to keep automatically decides which n − r to leave, so C(10, 3) and C(10, 7) both equal 120. The edges are tidy too: C(n, 0) and C(n, n) are both 1, since there is exactly one way to choose nothing and one way to choose everything. Read a large result as long odds — 13,983,816 lottery combinations means a single ticket has that-in-one odds of being the winning draw.
The combinations formula is exact, but a couple of points keep it from being misapplied.
Order off, no repeats, r ≤ n
This calculator counts combinations — selections where order does not matter and each item is chosen at most once. If order matters, you want permutations instead; if items may repeat, you need combinations with repetition, a different formula. Both n and r must be whole numbers, and you cannot choose more items than exist, so r must be no greater than n — otherwise there is no valid selection and the calculator returns no result.