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Difference between nCr and nPr

Last Updated : 23 Jul, 2025

When studying permutations and combinations in mathematics, two essential concepts often arise: nPr and nCr. These terms are used to describe different ways of selecting and arranging items from a larger set.

Permutations (nPr) refer to the number of ways to arrange r objects out of n total objects, where the order of arrangement matters. Combinations (nCr), on the other hand, refer to the number of ways to select r objects from n objects without regard to the order of selection. In this article, we will discuss the key differences between nPr and nCr.

What is nCr?

The term nCr (also written as ) refers to the number of combinations of n items taken r at a time without regard to the order of selection. It is also known as the binomial coefficient.

Let's say you have a set of 4 fruits: {Apple, Banana, Cherry, Date}, and you want to choose 2 fruits from this set.

There are 6 possible combinations of 2 fruits from the set of 4 fruits:

  • {Apple, Banana}
  • {Apple, Cherry}
  • {Apple, Date}
  • {Banana, Cherry}
  • {Banana, Date}
  • {Cherry, Date}

This can be calculated using the nCr, where n is 4 and r is 2.

Formula for nCr

The formula to calculate nCr is given by:

nCr =

Where n! (n factorial) is the product of all positive integers up to n.

Calculation for nCr

Let's calculate 5C2:

5C2 =

ā‡’ 5! = 5 Ɨ 4 Ɨ 3 Ɨ 2 Ɨ 1 = 120

ā‡’ 2! = 2 Ɨ 1 = 2

ā‡’ 3! = 3 Ɨ 2 Ɨ 1 = 6

So, 5C2 = 120/(2 Ɨ 6) = 120/12 = 10

Therefore, there are 10 ways to choose 2 items from a set of 5 items.

What is nPr?

The term nPr (also written as P(n, r) refers to the number of permutations of n items taken r at a time. In permutations, the order of selection matters, unlike in combinations where the order does not matter.

For example, if we are choosing 2 items from a set {A, B, C}, the permutations would be:

  • {A, B}
  • {B, A}
  • {A, C}
  • {C, A}
  • {B, C}
  • {C, B}

Notice that {A, B} and {B, A} are considered different permutations because the order is different.

Formula for nPr

The formula to calculate nPr is given by:

nPr = n!/(nāˆ’r)!

where n! (n factorial) is the product of all positive integers up to n.

Calculation for nPr

Let's calculate 5P2:

5P2 = 5!/(5 āˆ’ 2)! ā€‹= 5!/3!ā€‹

5! = 5 Ɨ 4 Ɨ 3 Ɨ 2 Ɨ 1 = 120, and 3! = 3 Ɨ 2 Ɨ 1 =6

So, 5P2 = 120/6 = 20

Therefore, there are 20 ways to arrange 2 items out of a set of 5 items in order.

Difference between nCr and nPr

The key differences between nCr and nPr are listed in the following table:

nCr or C(n, r) or

FeaturenCr(Combination)nPr (Permutation)
DefinitionThe number of ways to choose r elements from a set of n elements without regard to the order.The number of ways to arrange r elements from a set of n elements, considering the order.
FormulanCr = n!/[r!(nāˆ’r)!]P(n, r) = n!/(n āˆ’ r)!
OrderOrder does not matterOrder matters
Use CaseSelecting team members, forming groupsArranging books on a shelf, scheduling tasks
SynonymsCombinationPermutation
Mathematical NotationP(n, r) or nPr
Key CharacteristicsCombinations are subsetsPermutations are sequences

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