Combinations Calculator – Calculate nCr
Enter the total number of objects and the sample size to calculate the possible combinations with the n choose k combinations calculator.
Result:
Number of Combinations C(n,r)
Number of Permutations P(n,r)
On this page:
How to Calculate Combinations
Combinations are a set of items from a larger collection without regard to the order of the items in the set. Combinations are often represented as nCr or n choose r.
Lottery tickets are a common example to demonstrate what combinations are. Each ticket represents a combination of numbers from a larger set.
In fact, once you know the total number of combinations, then you can calculate the probability of winning. In fact, this is the basis for calculating probabilities in a binomial distribution.
The calculator above finds the number of possible combinations of things in a collection. You can also use a formula the calculate the number of combinations in a set.
Combinations Formula
The following formula defines the number of possible combinations of r items in a collection of n total items.
C(n,r) = n!r!(n – r)!
Thus the number of possible combinations of r items in a set of n items is equal to n factorial divided by r factorial times n minus r factorial.
N Choose K
Combinations are also sometimes referred to as n choose k where n is the total number of items and k is the number of items in the sample. Thus the number of combinations is equal to the number of ways you can choose k items from a set of n total items.
N Choose K Formula
The n choose k formula is nearly identical to the combinations formula above, but the number of items in the sample is denoted k, and the total number of objects is denoted n.
C(n,k) = n!k!(n – k)!
Combinations vs. Permutations
Combinations are very similar to permutations with one key difference: the number of permutations is the number of ways to choose r objects in a set of n objects in a unique order. Thus, with permutations, the order of the objects in the set is important.
In most cases, there will be more possible permutations of objects in a set because each different order of the items is considered a different permutation, but with combinations, the order is irrelevant, and those are considered the same combination of items.
For example, see the combinations and permutations of 2 letters in a set of the letters {A, B, C}.
Combinations (3)
(A, B)
(A, C)
(B, C)
Permutations (6)
(A, B)
(B, A)
(A, C)
(C, A)
(B, C)
(C, B)
Thus, for a sample of 2 objects in a total number of 3 objects, there are 3 combinations and 6 permutations.
Permutations Formula
The formula below states how to calculate permutations:
P(n,r) = n!(n – r)!
Thus the number of possible permutations of r items in a set of n items is equal to n factorial divided by n minus r factorial.
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