Permutation Calculator

Q: How does word mode work?

It counts repeated letters and computes n!/(a!b!c!...) exactly using BigInt.

Calculate permutations and combinations with exact results (BigInt). Use the Order matters? toggle to automatically choose P(n,k) vs C(n,k). Also includes word permutations (multiset) for cases like “MISSISSIPPI”.

Background

If order matters, you’re counting permutations. If order doesn’t matter, you’re counting combinations. This calculator routes you to the correct formula automatically.

How to use this calculator

Set Order matters? to choose permutations vs combinations automatically.
Enter n and k (or type a word for repeated letters).
Click Calculate for an exact BigInt answer (no rounding).
Turn on Steps to see formulas and exact computation details.

How this calculator works

Validates inputs (integers, non-negative, and logical constraints like 0 ≤ k ≤ n when needed).
If Order matters = Yes, uses permutations. If No, uses combinations.
For word mode, counts repeated letters and computes multiset permutations exactly.
Uses BigInt to avoid overflow and rounding.

Formula & Equation Used

Permutations (no repetition): P(n,k) = n! / (n-k)!

Combinations: C(n,k) = n! / (k!(n-k)!)

With repetition (ordered): n^k

Combinations with repetition (unordered): C(n+k-1, k)

Multiset / word permutations: n! / (a! b! c! ...)

Example Problems & Step-by-Step Solutions

Example 1 — Order matters (10 P 3)

Order matters → permutations.
P(10,3)=10×9×8=720

Example 2 — Order doesn’t matter (10 C 3)

Order doesn’t matter → combinations.
C(10,3)=120

Example 3 — Word permutations (MISSISSIPPI)

Counts: M=1, I=4, S=4, P=2 → total n=11.
11! / (1! 4! 4! 2!) (exact)

Frequently Asked Questions

Q: How do I know if order matters?

If swapping positions creates a different outcome (1st vs 2nd), order matters → permutations.

Q: What if order doesn’t matter?

Use combinations: choosing a group where only the members matter, not the order.