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)

 
Learn how we calculated this below


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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.

formula showing that the number of combinations is equal to n factorial divided by r factorial times n minus r factorial
The combinations formula.

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)!

n choose k formula showing that the number of combinations is equal to n factorial divided by k factorial times n minus k factorial
The n choose k formula is also often represented as an n over the k in parentheses.

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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