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  4. Permutation and Combination Calculator

Permutation and Combination Calculator

nPr and nCr without replacement, plus with-replacement n^r and multichoose, with factorials and steps.

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How it works

Order matters for permutations; combinations ignore order. Without replacement is the standard nPr/nCr pair.

nPr=n!/(n−r)! · nCr=n!/(r!(n−r)!) · n^r · C(n+r−1,r)

When to use

Use this tool when you need to: calc.net demo: n=6, r=2 → P=30, C=15. Soccer: 11P2=110 captains/keepers; 11C2=55 pairs of strikers.

How to use this tool

  1. Enter your values in the fields above.
  2. Review the formula and any mode options for your problem.
  3. Read the result and the step-by-step solution.

Calc.net demo: n=6, r=2 → P=30, C=15. Soccer: 11P2=110 captains/keepers; 11C2=55 pairs of strikers.

Frequently asked questions

What is the Permutation and Combination Calculator?
Order matters for permutations; combinations ignore order. Without replacement is the standard nPr/nCr pair.
How do I use the Permutation and Combination Calculator?
Enter the required values, then calculate. The Permutation and Combination Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Permutation and Combination Calculator use?
nPr=n!/(n−r)! · nCr=n!/(r!(n−r)!) · n^r · C(n+r−1,r)

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Permutation and combination calculator

Enter total set size n and subset size r. Get nPr and nCr without replacement, plus the with-replacement formulas n^r and C(n+r−1, r).

Integers with 0 ≤ r ≤ n ≤ 5000. Primary nPr / nCr never reuse an element. Replacement formulas allow repeats (e.g. 3-3-3).

Try an example

How it works

Permutations vs combinations

Both live in combinatorics — counting finite structures. Permutations care about order; combinations do not. A padlock that needs 1-2-9 vs 2-9-1 is a permutation lock, not a “combination” lock in the strict sense. The main results here are without replacement (no value reused after it is chosen).

Permutations — nPr

r-permutations of n: ordered selections of r distinct items from n. Formula nPr = n! / (n−r)! = n×(n−1)×…×(n−r+1). Example: captain then keeper from 11 players → 11P2 = 11×10 = 110. With replacement (ordered, repeats OK) the count is simply n^r.

Combinations — nCr

Unordered selections of r distinct items. Divide out the r! orderings of each group: nCr = n! / (r!(n−r)!) = nPr / r!, also written as the binomial coefficient C(n,r). Example: two strikers from 11 → 11C2 = 55. With replacement (multisets) use C(n+r−1, r).

With vs without replacement

Without replacement: each drawn item is removed (team roles, lottery numbers). With replacement: the same value can appear again (some codes allow 3-3-3). This calculator always reports both so you can compare.

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