Calculator Wizard
What Is a Combination?
A combination A selection of items where order does not matter. Picking A then B is the same combination as picking B then A. counts the number of ways to choose r items from a set of n, where the order you pick them in doesn't matter. Once a group is chosen, there's no "1st pick" or "2nd pick" distinction — just who's in the group.
The extra r! in the denominator (compared to a permutation) exists specifically to cancel out all the different orderings of the same group, since combinations don't care about order.
Order Matters vs. Doesn't — The Core Distinction
This is genuinely the single most important thing to get right before using this calculator. Combinations (nCr) count selections where order doesn't matter — like choosing 3 people for a committee, where there's no distinction between who was "picked first." Its close relative, permutation An arrangement of items where the order matters. Swapping two items creates a different, distinct permutation. (nPr), counts arrangements where order does matter — like awarding 1st, 2nd, and 3rd place, where the same three people in a different order is a genuinely different outcome.
| Combination (nCr) | Permutation (nPr) | |
|---|---|---|
| Does order matter? | No | Yes |
| Formula | n! ÷ (r! × (n − r)!) | n! ÷ (n − r)! |
| Example scenario | Choosing 3 people for a committee | Assigning 1st/2nd/3rd place medals |
| Result size | Always smaller (or equal) | Always larger (or equal) |
Worked Example
Example: Choosing 3 people for a committee out of 5 candidates
Step 1: n = 5, r = 3
Step 2: nCr = 5! ÷ (3! × (5 − 3)!) = 120 ÷ (6 × 2)
Step 3: nCr = 10
There are 10 different possible 3-person committees from 5 candidates — because a committee of Alice, Bob, and Carol is the same committee no matter which order they were selected in.
How is nCr related to nPr?
nCr is always nPr divided by r! (r factorial) — since a combination just ignores the different orderings a permutation would count separately. For the example above: 5P3 = 60, and 60 ÷ 3! (which is 6) = 10, matching the nCr result exactly.
What if r is larger than n?
That's not mathematically valid — you can't choose more items than you have available. Make sure r is less than or equal to n.
Need order to matter instead?
Try the Permutations Calculator (nPr)