Permutation & Combination

Calculate permutations and combinations for probability

About Permutation & Combination

Permutations and combinations both count the number of ways to select items from a group, but permutations count arrangements where order matters, while combinations count selections where order doesn't matter. They're foundational to probability, statistics, and combinatorics problems.

Formula

Permutations: nPr = n! / (n − r)! | Combinations: nCr = n! / (r! × (n − r)!)

How It Works

Enter the total number of items (n) and how many you're selecting or arranging (r)
Choose whether order matters (permutation) or doesn't matter (combination)
The calculator applies the matching factorial formula
Result shows the total number of possible permutations or combinations

Tips

If you're arranging items in a specific order (like race finishing positions), use permutations
If you're just selecting a group without regard to order (like choosing a committee), use combinations
nCr is always smaller than or equal to nPr for the same n and r, since combinations group together all the different orderings of the same selection

Frequently Asked Questions

What's the difference between a permutation and a combination?

A permutation counts arrangements where order matters (like 1st, 2nd, 3rd place); a combination counts selections where order doesn't matter (like picking 3 people for a team) — the same group counted as multiple permutations collapses into a single combination.

How many ways can I arrange 5 items?

5! (5 factorial) = 5 × 4 × 3 × 2 × 1 = 120 total arrangements, since this is a permutation of all 5 items taken 5 at a time.

How do I calculate 'n choose r' (combinations)?

nCr = n! / (r! × (n−r)!). For example, choosing 3 people from a group of 8: 8! / (3! × 5!) = 56 possible combinations.

When should I use permutations instead of combinations?

Use permutations whenever the order or arrangement matters to the outcome — like assigning distinct roles, ranking results, or forming passwords — and combinations when you're only choosing a group or subset with no ranking involved.

Why is 0! (zero factorial) equal to 1?

By mathematical convention, 0! is defined as 1 so that formulas like nPr and nCr work correctly even when there's exactly one way to arrange zero items (doing nothing).