Math & Stats

Permutation and Combination Calculator

Calculate permutations (nPr) and combinations (nCr) instantly. Enter n and r to find the number of ordered and unordered selections using exact BigInt math.

Permutation and Combination Calculator

Total items (n)

Items chosen (r)

Method

How this calculator works

Permutations: nPr = n! / (n − r)! counts ordered selections. Combinations: nCr = n! / (r! × (n − r)!) counts unordered selections. Both require 0 ≤ r ≤ n.

Enter the total number of items, n (for example 10).
Enter how many you are selecting, r (for example 3), making sure r ≤ n.
Click Calculate to see both nPr (ordered) and nCr (unordered) results.

Examples

Worked examples

Real numbers, end-to-end results.

10 P 3

= 720

Ordered selections of 3 from 10: 10 × 9 × 8.

10 C 3

= 120

Unordered groups of 3 from 10: 720 ÷ 3!.

5 C 5

= 1

There is exactly one way to choose all items.

Use cases

When to use it

Finding the odds of lottery or raffle draws (combinations).
Counting possible passwords, PINs, or race finishing orders (permutations).
Calculating binomial coefficients for probability and statistics.
Planning committee, team, or seating selections.

FAQ

Frequently asked questions

What is the difference between a permutation and a combination?

A permutation counts arrangements where order matters (ABC differs from CAB), while a combination counts selections where order does not matter (ABC is the same group as CAB). That is why nPr is always greater than or equal to nCr.

What do n and r represent?

n is the total number of items available to choose from, and r is the number of items you are selecting. Both must be non-negative integers, and r cannot exceed n.

What is nP0 or nC0?

Both equal 1. There is exactly one way to select nothing from a set — the empty selection — so nP0 = nC0 = 1 for any valid n.

Why use BigInt for these calculations?

Permutations and combinations involve factorials, which grow extremely large very quickly. BigInt arithmetic keeps the results exact even for big values of n, avoiding the rounding errors of floating-point math.

Can r be larger than n?

No. You cannot select more items than are available, so r must satisfy 0 ≤ r ≤ n. If you enter r greater than n, the calculator shows an error.