Combinations Calculator
Unordered selections.
Popular n choose r values
What does Combinations mean?
A combination — written nCr and read aloud as “n choose r” — is a way of selecting r items from a set of n where the order of selection does not matter. It answers questions like: how many ways can I choose 3 students from a class of 10 to form a committee? That is 10 choose 3, and the answer is 120. Combinations are fundamental in probability, statistics, and combinatorics, and appear in everyday scenarios like lottery odds (49 choose 6), card hands (52 choose 5), team formation, and menu selections.
How to calculate Combinations
n choose r is calculated with the formula C(n, r) = n! / (r! × (n − r)!), where n is the total number of items and r is the number of items to choose. For example, 10 choose 3 is C(10, 3) = 10! / (3! × 7!) = 720 / 6 = 120. For comparison, the permutation formula P(n, r) = n! / (n − r)! counts ordered arrangements — P(10, 3) = 720, which is always greater than or equal to the combination count.
FAQ
Combinations count selections where order does not matter (e.g., choosing team members), while permutations count arrangements where order matters (e.g., assigning 1st, 2nd, 3rd place). For the same n and r, permutations are always greater than or equal to combinations because each combination can be arranged in r! different orders.
Use combinations when the order of selection is irrelevant — for example, choosing a committee, selecting lottery numbers, or picking toppings for a pizza. Use permutations when the arrangement or sequence matters — such as ranking contestants, assigning seats, or creating passwords.
Yes. If r = 0, there is exactly one way to choose nothing: C(n, 0) = 1. If n = 0 and r = 0, the result is also 1 by convention (0! = 1). However, r cannot be greater than n because you cannot choose more items than are available.
Lottery odds (e.g., choosing 6 numbers from 49), poker hands (5 cards from 52), forming committees or teams, selecting toppings from a menu, and choosing subjects to study from a list of electives are all everyday examples of combinations.
Factorials grow extremely fast. For standard floating-point numbers, n! overflows around n = 170. For most practical purposes with this calculator, values of n up to about 170 will produce accurate results. For larger values, specialized big-integer libraries are needed.
52 choose 5 is C(52, 5) = 52! / (5! × 47!) = 2,598,960 — the number of distinct 5-card hands you can deal from a standard deck. Enter n = 52 and r = 5 above to see it, or work through any other pair the same way: divide n! by r! × (n − r)!.
nCr stands for "n choose r" and is the standard notation for combinations. The "n" represents the total number of items, "C" stands for combinations, and "r" is the number of items being chosen. For example, 10C3 (also written as C(10,3) or ₁₀C₃) means "how many ways can I choose 3 items from 10?" The answer is 120. nCr is widely used in probability, statistics, and combinatorial mathematics.
Related calculators
- Permutations— Ordered arrangements.
- Factorial— Product of descending integers.
- Probability— Likelihood of outcome.
- Ratio— Relative comparison of values.