Calculate the number of ways to choose r items from n items without regard to order using the nCr formula.
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Combinations (nCr)
There are 120 ways to choose 3 items from 10 without regard to order.
Combinations vs. permutations
A combination counts selections where order does not matter. By contrast, the 720 permutations count arrangements where order does matter. The combination value is always less than or equal to the permutation value because multiple orderings of the same selection collapse into one combination.
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Combinatorics
A combination calculator finds the number of ways to select r items from a set of n items when the order of selection does not matter. It uses the nCr formula, also written as C(n,r) or "n choose r", which is fundamental to probability, statistics, and discrete mathematics.
A combination counts selections where order is irrelevant. Choosing players A, B, C for a team is the same selection as C, A, B. The formula divides the total permutations by r! to remove duplicate orderings.
For example, C(52,5) calculates how many different 5-card poker hands can be dealt from a standard 52-card deck. The answer is 2,598,960.
C(n, r) = n! / (r! × (n − r)!)
The number of unordered selections of r items from n items.
Permutations count ordered arrangements (ABC ≠ BAC), while combinations count unordered selections (ABC = BAC). Every combination of r items corresponds to r! permutations, so C(n,r) = P(n,r) / r!. Use combinations for team selection, lottery odds, and committee formation. Use permutations for rankings, passwords, and sequences.
For C(10,3): there are 120 combinations but 720 permutations, because each group of 3 can be arranged in 3! = 6 different orders.
Frequently asked questions
"n choose r" means the number of ways to select r items from n items without regard to order. It is written as C(n,r), nCr, or with the binomial coefficient notation. The result is always a positive integer.
Both equal 1. There is exactly one way to choose nothing (the empty set) and exactly one way to choose everything (the full set).
Combinations calculate the number of equally likely outcomes in many probability problems. For example, the probability of a specific poker hand is the number of ways to get that hand divided by C(52,5) = 2,598,960 total possible hands.
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