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Permutations Combinations Calculator

Permutations & Combinations

Instantly calculate nPr and nCr with our advanced 2026 calculator. Get step-by-step solutions for your probability and statistics problems.

PPermutations (nPr)

Order matters. Arranging r items from a set of n.

Result
720

Calculation Steps

1.Formula: P(n, r) = n! / (n - r)!
2.Substitute values: P(10, 3) = 10! / (10 - 3)!
3.Simplify denominator: P(10, 3) = 10! / 7!
4.Expand: 10 × 9 × 8
5.Final Result: 720

CCombinations (nCr)

Order doesn't matter. Choosing r items from a set of n.

Result
120

Calculation Steps

1.Formula: C(n, r) = n! / (r! × (n - r)!)
2.Substitute values: C(10, 3) = 10! / (3! × (10 - 3)!)
3.Simplify: C(10, 3) = 10! / (3! × 7!)
4.Calculate Permutations (Numerator): 720
5.Calculate Factorial (Denominator): 6
6.Divide: 720 / 6
7.Final Result: 120

Understanding Permutations vs. Combinations

In combinatorics, it is crucial to distinguish between Permutations and Combinations. Both concepts involve selecting items from a group, but the deciding factor is order.

Permutations (nPr)

Order Matters. Use this when the sequence or arrangement of items is important.

  • ✅ Combination locks
  • ✅ Race podiums (1st, 2nd, 3rd)
  • ✅ URL slugs

Combinations (nCr)

Order Does Not Matter. Use this when you just need to know which items are selected, regardless of sequence.

  • ✅ Lottery numbers
  • ✅ Card hands (Poker)
  • ✅ Pizza toppings

Permutation Formula (nPr)

The number of ways to arrange r items from a set of n distinct items is given by:

P(n, r) = n! / (n - r)!

Where n! (n factorial) is the product of all positive integers less than or equal to n. To calculate factorials, use our Factorial Calculator.

Combination Formula (nCr)

The number of ways to choose r items from a set of n distinct items is given by:

C(n, r) = n! / [r! × (n - r)!]

Notice the extra r! in the denominator. This removes the duplicates that occur when order doesn't matter (e.g., "A, B" is the same as "B, A").

Comparison Table

FeaturePermutations (nPr)Combinations (nCr)
OrderMattersDoes Not Matter
KeywordsArrange, Order, SequenceSelect, Choose, Group
MagnitudeHigh (More possibilities)Low (Fewer possibilities)

Related Math Calculators

Permutations and combinations are built on factorials and are related to other mathematical concepts:

Factorial Calculator →
Calculate n! (factorial) which is essential for permutations and combinations formulas

Frequently Asked Questions

What is the difference between permutations and combinations?
The key difference is order. In permutations (nPr), the order of arrangement matters (e.g., a combination lock code). In combinations (nCr), the order does not matter (e.g., lottery numbers). If swapping two items changes the outcome, use permutations.
How do I calculate permutations (nPr)?
The formula for permutations is nPr = n! / (n - r)!. This calculates the number of ways to arrange 'r' distinct items from a set of 'n' items.
How do I calculate combinations (nCr)?
The formula for combinations is nCr = n! / (r! * (n - r)!). It acts like the permutation formula but divides by r! to remove duplicate orderings, as order does not matter.
Can nPr be smaller than nCr?
No. Permutations (nPr) are always greater than or equal to Combinations (nCr) for the same n and r (where r > 0), because permutations count every specific order as unique.