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.

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.

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.