Permutation & Combination Calculator (nPr & nCr)

Permutation & Combination Calculator – nPr & nCr
n must be a whole number ≥ 0
r must be a whole number ≥ 0

Results

Enter values for n and r.
nPr = n!/(n−r)!, nCr = n!/[r!(n−r)!]

Permutation & Combination Calculator (nPr & nCr)

This free tool calculates permutations (nPr) and combinations (nCr) instantly. Enter values for n (total items) and r (chosen items), and get results with full formula steps.

Permutations and Combinations Explained

In mathematics and probability, permutations and combinations are two ways of counting selections:

  • Permutation (nPr): The number of ways to arrange r items out of n when order matters.
  • Combination (nCr): The number of ways to choose r items out of n when order does not matter.

Example:

  • Arranging 3 books out of 5 on a shelf → Permutation.
  • Choosing 3 lottery numbers out of 5 → Combination.

Permutation Formula (nPr)

nPr = n! / (n − r)!

Example: With 6 players and 3 prizes to award:

6P3 = 6! / (6 − 3)! 
    = 6 × 5 × 4 
    = 120

Combination Formula (nCr)

nCr = n! / [ r! × (n − r)! ]

Example: Choosing 2 fruits from 4:

4C2 = 4! / (2! × (4 − 2)!) 
    = (4 × 3) / (2 × 1) 
    = 6

How to Use the Calculator

  1. Enter n (total items).
  2. Enter r (items to choose).
  3. Select whether to calculate Permutation (nPr), Combination (nCr), or both.
  4. (Optional) Check “Combination with Repetition” to use the formula: C(n + r − 1, r).
  5. Click Calculate to see instant results and step-by-step working.

Real-World Applications

  • Permutations (nPr): seating arrangements, passwords, race outcomes, order-sensitive scenarios.
  • Combinations (nCr): lottery draws, team selection, card games, probability without order.
  • Combination with Repetition: ice cream flavours (choose scoops with repeats allowed).

FAQs

Q: What is the difference between permutation and combination?

  • Permutation: order matters.

  • Combination: order doesn’t matter.

Q: Can nPr or nCr be negative?
No. Both require n ≥ 0, r ≥ 0, and r ≤ n (unless using combinations with repetition).

Q: Why do formulas use factorials?
Factorials simplify counting by multiplying consecutive integers. That’s why both nPr and nCr are built on factorials.

Q: Where are these formulas used in real life?
Probability, statistics, cryptography, seating arrangements, computer science, and more.

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