Permutation & Combination Calculator · Practical guide
Permutation vs. Combination: When Order Matters
One question decides it: does rearranging the same items count as a different outcome?
At a glance
The One-Question Test
Ask: does the order of the chosen items matter? If yes, use a permutation: nPr = n! ÷ (n − r)!. If no, use a combination: nCr = n! ÷ (r! × (n − r)!). Picking a president, vice president, and treasurer from 10 people is a permutation — the same three people in different roles are different outcomes. Picking a 3-person committee is a combination.
Interactive tool
Permutation & Combination Calculator
Enter your own values, then compare the result with the examples below.
Results
Formulas:
nPr = n! / (n - r)!
nCr = n! / (r! × (n - r)!)
Side by Side: n = 10, r = 3
| Permutation P(10,3) | Combination C(10,3) | |
|---|---|---|
| Order matters? | Yes | No |
| Formula | 10! ÷ 7! | 10! ÷ (3! × 7!) |
| Result | 720 | 120 |
| Example | President, VP, treasurer | 3-person committee |
The two answers always differ by exactly r! — here 3! = 6 — because every unordered group can be arranged that many ways. Formally, nPr = nCr × r!.
Everyday Examples
- Combination: lottery numbers, pizza toppings, handshakes at a meeting, books to pack for a trip.
- Permutation: lock codes, race finishing order, seating arrangements, rankings and podiums.
A 3-digit code using distinct digits from 0–9 is P(10,3) = 720 possible codes. A 6-number lottery pick from 49 is C(49,6) = 13,983,816 combinations — which is why jackpots are rare.
Frequently Asked Questions
What is a quick way to decide which one to use?
Ask whether rearranging the same items produces a new outcome. Gold-silver-bronze is different from silver-gold-bronze (permutation); a committee of Alice, Bob, and Carol is the same in any order (combination).
Is a lottery a permutation or a combination?
A combination, because the order the numbers are drawn does not matter — only the set of numbers on your ticket does.
Are handshake problems combinations?
Yes. A handshake between A and B is the same as between B and A, so n people shake hands C(n,2) times — 10 people, 45 handshakes.
Can nCr ever be larger than nPr?
Never. Since nPr = nCr × r! and r! is always at least 1, permutations are always greater than or equal to combinations for the same n and r.