About the Permutation and Combination
This calculator counts arrangements and selections: permutations, where order matters, and combinations, where it does not. These two counts underpin probability, lottery odds, password strength, and any problem that asks how many ways something can be done.
The formula
P(n,r) = n! / (n − r)! C(n,r) = n! / (r!(n − r)!)n is the number of items available and r the number chosen. The difference is the r! in the combination denominator, which divides out the orderings of each selection.
How to use this calculator
- 1Enter your Total Items (n). The field starts at
10, which you can overwrite. - 2Enter your Items Chosen (r). The field starts at
3, which you can overwrite. - 3Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Total Items (n) | 10 |
| Items Chosen (r) | 3 |
Result
P(10,3) = 720
C(10,3) = 120
120 unique groups
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
Order is the whole distinction. Choosing three people from ten for gold, silver, and bronze medals is a permutation — 720 outcomes, since who gets which medal matters. Choosing three for an unranked committee is a combination — 120 outcomes, because the same three people are one committee however you list them. The ratio between the two is exactly 3! = 6, the number of ways to order three items.
Combinations are always fewer than permutations, and both grow fast. This is why brute-force approaches fail so quickly, and why lottery odds are so long: choosing 6 numbers from 49 gives nearly 14 million combinations, so a single ticket has that chance of matching.
Things worth knowing
- Ask whether rearranging the same selection counts as different. If yes, permutation; if no, combination.
- C(n, r) equals C(n, n − r) — choosing 3 from 10 to include is the same count as choosing 7 to exclude.
- These formulas assume selection without replacement. If items can repeat, different formulas apply.
- A lock combination is really a permutation, since the order of digits matters. The everyday name is wrong.
- C(n, 0) and C(n, n) both equal 1: one way to choose nothing, one way to choose everything.
Frequently asked questions
What is the difference between a permutation and a combination?+
Permutations count arrangements where order matters; combinations count selections where it does not. Three medal winners from ten is a permutation; three committee members from ten is a combination.
How do I know which one to use?+
Ask whether reordering the same items creates a different outcome. Podium positions and PIN codes are permutations; lottery tickets and committee memberships are combinations.
How are lottery odds calculated?+
With combinations, since draw order does not matter. Choosing 6 from 49 gives C(49,6) = 13,983,816 possible tickets, so one ticket has a 1 in 13,983,816 chance of matching all six.
Why do permutations outnumber combinations?+
Because each combination can be arranged in r! different orders, and permutations count all of them separately. That factor of r! is exactly the difference between the two formulas.