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.

Table of Content

What is nCr?

The term nCr (also written as \binom{n}{r}) 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 = \frac{n!}{r!(n-r)!}

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

Calculation for nCr

Let's calculate 5C2:

5C2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!}

⇒ 5! = 5 × 4 × 3 × 2 × 1 = 120

⇒ 2! = 2 × 1 = 2

⇒ 3! = 3 × 2 × 1 = 6

So, 5C2 = 120/(2 × 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!/(n−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 − 2)! ​= 5!/3!​

5! = 5 × 4 × 3 × 2 × 1 = 120, and 3! = 3 × 2 × 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 \binom{n}{r}

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!(n−r)!]P(n, r) = n!/(n − 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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