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nCr
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Mathematics • Combinatorics

Permutation & Combination Calculator

Calculate nPr and nCr with precision, detailed formulas, and interactive steps.

Advanced support for 5C3, 10P4, repetition, circular, and multiset modes.

Exact n=200 Detailed Steps Advanced Modes Mode: Standard
Formula reference
Permutation
P(n, r) = n! (n - r)!
Combination
C(n, r) = n! r! × (n - r)!
Example: 10P3 = 720, 10C3 = 120

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Advanced Calculation Modes

Supports nCr, nPr, 5C3, 6 choose 3, 4P3, C(5,3), and P(10,4).

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Permutation
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Enter values for n and r to see formulas, steps, and the final answer.

Example: 8C3 = 56 Example: 7P2 = 42

Free Permutation and Combination Calculator

Our online permutations and combinations calculator helps you calculate nPr (permutations) and nCr (combinations) instantly. Whether you need to calculate 5C3, 6 choose 3, 5C4, or any other combination or permutation value, this free calculator provides accurate results with step-by-step explanations and formulas.

Common Calculations - Quick Reference

Combinations (nCr):
  • 5C3 (5 choose 3) = 10
  • 6C3 (6 choose 3) = 20
  • 5C4 (5 choose 4) = 5
  • 6C2 (6 choose 2) = 15
  • 5C2 (5 choose 2) = 10
Permutations (nPr):
  • 5P3 (5 permute 3) = 60
  • 6P3 (6 permute 3) = 120
  • 4P3 (4 permute 3) = 24
  • 3P2 (3 permute 2) = 6
  • 2P4 requires n≥r

How to Calculate Permutations and Combinations

Permutations (nPr): Use when order matters. Formula: P(n,r) = n! / (n-r)!
Example: Selecting 1st, 2nd, 3rd place from 10 people = 10P3 = 720

Combinations (nCr): Use when order does not matter. Formula: C(n,r) = n! / (r! × (n-r)!)
Example: Choosing 3 team members from 10 people = 10C3 = 120

Tip: Permutations are always greater than or equal to combinations for the same n and r values.

Understanding Permutations & Combinations

Permutation (nPr)

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

When to use: When the order of selection matters

Example: Selecting 3 winners (1st, 2nd, 3rd) from 10 contestants - order matters because positions are different

Combination (nCr)

Formula: C(n, r) = n! / (r! × (n - r)!)

When to use: When the order of selection does not matter

Example: Selecting 3 team members from 10 people - order does not matter because all members have equal roles

Key Points

  • n must be greater than or equal to r (n ≥ r)
  • Both n and r must be non-negative integers
  • P(n, r) ≥ C(n, r) - Permutations are always greater than or equal to combinations
  • Factorial (!) means: n! = n × (n-1) × (n-2) × ... × 2 × 1
  • Special case: 0! = 1

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