Permutations vs Combinations: When Order Matters
The core distinction: permutations count arrangements where order matters, while combinations count selections where order does not matter. Choosing the wrong one will give you an answer that is off by a factor of many thousands.
| Dimension | Permutation (nPr) | Combination (nCr) |
|---|---|---|
| Definition | Ordered arrangements of r items from n | Unordered selections of r items from n |
| Formula | n! / (n − r)! | n! / [r! × (n − r)!] |
| Order matters? | Yes | No |
| Result size | Always ≥ nCr | Smaller or equal to nPr |
| Key symbol | P(n,r) or nPr | C(n,r), nCr, or "n choose r" |
What Are Permutations?
A permutation is an ordered arrangement. When you arrange 3 books from a shelf of 10, the order in which the books appear matters: "A, B, C" is a different arrangement from "C, A, B." The number of such arrangements is:
nPr = n! / (n − r)!
For 10 books taken 3 at a time: 10! / 7! = 720 arrangements. Classic permutation problems include PIN codes, race finishing orders, and seating arrangements. Use the Permutation Calculator (nPr) to compute any nPr instantly.
What Are Combinations?
A combination is an unordered selection. When you choose 3 toppings from 10 for a pizza, the toppings you pick matter but not the order you pick them: "olives, peppers, mushrooms" is the same selection as "mushrooms, olives, peppers." The count is:
nCr = n! / [r! × (n − r)!]
For 10 toppings taken 3 at a time: 10! / (3! × 7!) = 120 selections. Classic combination problems include lottery draws, committee formation, and card hands. Use the Combination Calculator (nCr) to compute any nCr without doing factorial arithmetic by hand.
Key Differences
Every combination corresponds to r! different permutations (all the ways to arrange that selection). This gives the relationship:
nPr = nCr × r!
For the example above: 720 = 120 × 6. Permutations are always at least as large as combinations; they are equal only when r = 0 or r = 1.
- Permutations: lock combinations (despite the misleading name), ranked lists, scheduling ordered tasks.
- Combinations: lottery tickets, choosing a committee, dealing a poker hand.
Which Should You Use?
Ask one question: Does swapping two items create a new outcome?
- If yes -- use permutations. A PIN of 1-2-3-4 is different from 4-3-2-1.
- If no -- use combinations. A committee of Alice, Bob, Carol is the same regardless of the order you named them.
When in doubt, work a tiny example with 2 or 3 items and count by hand. If your manual count matches nCr, use combinations; if it matches nPr, use permutations.
The Permutation Calculator (nPr) and the Combination Calculator (nCr) both show intermediate factorial steps so you can verify your reasoning.
FAQ
Why is a combination lock not actually a combination?
In everyday speech, "combination lock" is a misnomer. The sequence 3-7-1 is different from 1-7-3, so order matters and the correct term is a permutation lock.
What if repetition is allowed?
With repetition, permutations become nr and combinations use a different formula (n + r − 1)! / [r! × (n − 1)!]. The calculators linked above handle the standard no-repetition case.
How do permutations and combinations relate to probability?
Many probability problems divide a favorable count by a total count, both computed with the same method. Use combinations when the sample space is unordered (e.g., lottery) and permutations when it is ordered (e.g., race outcomes).