Calculator Wizard
What Is a Permutation?
A permutation An arrangement of items where the order matters. Swapping two items creates a different, distinct permutation. counts the number of ways to arrange r items chosen from a set of n, where the order they're arranged in matters. First place, second place, and third place are all different outcomes — even with the exact same three items.
Where n! ("n factorial") means multiplying n × (n−1) × (n−2) × ... down to 1.
Order Matters vs. Doesn't — The Core Distinction
This is genuinely the single most important thing to get right before using this calculator. Permutations (nPr) count arrangements where order matters — like 1st, 2nd, and 3rd place in a race. Its close relative, combination A selection of items where order does not matter. Picking A then B is the same combination as picking B then A. (nCr), counts selections where order doesn't matter — like picking a group of 3 people for a team, where there's no "1st pick" distinction once the group is chosen.
| Permutation (nPr) | Combination (nCr) | |
|---|---|---|
| Does order matter? | Yes | No |
| Formula | n! ÷ (n − r)! | n! ÷ (r! × (n − r)!) |
| Example scenario | Assigning 1st/2nd/3rd place medals | Choosing 3 people for a committee |
| Result size | Always larger (or equal) | Always smaller (or equal) |
Worked Example
Example: Awarding 1st, 2nd, and 3rd place among 5 finalists
Step 1: n = 5, r = 3
Step 2: nPr = 5! ÷ (5 − 3)! = 120 ÷ 2
Step 3: nPr = 60
There are 60 different ways to award gold, silver, and bronze among 5 finalists — because giving the gold to Finalist A instead of Finalist B is a genuinely different outcome, even if the same 3 people end up on the podium.
How is nPr related to nCr?
nPr is always nCr multiplied by r! (r factorial) — since a permutation is just a combination where you then separately arrange the chosen items in order. For the example above: 5C3 = 10, and 10 × 3! (which is 6) = 60, matching the nPr result exactly.
What if r is larger than n?
That's not mathematically valid — you can't arrange more items than you have available to choose from. Make sure r is less than or equal to n.
Need order to not matter instead?
Try the Combinations Calculator (nCr)