What's the difference between permutations and combinations?

Permutations count arrangements where order matters; combinations count selections where order doesn't matter.

How is the number of permutations calculated?

nPr = n! ÷ (n−r)!, for choosing r items from n in order.

How is the number of combinations calculated?

nCr = n! ÷ (r! × (n−r)!), for choosing r items from n regardless of order.

Permutation & Combination Calculator

Instantly calculate nPr and nCr, and understand the difference between them.

Total Items (n)

Items Chosen (r)

Permutations (nPr) — order matters-

Combinations (nCr) — order doesn't matter-

What Is a Permutation & Combination Calculator?

Permutations and combinations both count the ways to choose r items from a group of n, but they differ in whether order matters. A permutation counts arrangements where order matters (like a race's 1st, 2nd, 3rd place), while a combination counts selections where order doesn't matter (like picking a 3-person committee).

Order matters for permutations; it doesn't for combinations.

Formulas

nPr = n! ÷ (n − r)!
nCr = n! ÷ (r! × (n − r)!)

How to Use This Calculator

Enter the total number of items in your group (n).
Enter how many items you're choosing or arranging (r).
Click "Calculate" to see both the permutation count and combination count.

Worked Example

Example: Choosing 3 people from a group of 10 (n=10, r=3). Permutations = 10!/(10-3)! = 720 (ways to award 1st/2nd/3rd). Combinations = 720/3! = 120 (ways to just pick 3 people, no ranking).