Combination and Permutation Calculator

1. Enter the total items, n.
2. Enter how many you choose, r, which cannot exceed n.
3. Read the number of combinations, where order does not matter.
4. See the number of permutations, where order does.

Here is what each result means:

Combinations and permutations both count ways to choose r items from a group of n, but they differ on order. A combination ignores order, so picking apple then banana is the same as banana then apple. A permutation counts order, so those two orders are different. That single difference is why permutations are always at least as many as combinations.

Choosing 2 from 5 gives 10 combinations but 20 permutations, because each pair can be arranged in 2 orders. These counts are the backbone of probability, from lottery odds to card hands, and they build directly on the factorial.

The word combination in everyday speech, as in a combination lock, actually describes a permutation, since the order of the digits matters. In mathematics the two words are kept strictly apart: order matters for a permutation and never for a combination. Keeping that distinction clear is the key to choosing the right count.

It uses the standard formulas, both built from factorials.

1. Compute the permutations as n! divided by (n minus r)!.
2. Divide by r! to get the combinations, removing the orderings.
3. Report both exactly with big-integer arithmetic.

Both formulas rest on the factorial; the factorial calculator computes the pieces they use. Because the counts can grow into very long numbers, the tool works with big integers so that even a large lottery or card problem stays exact rather than rounding.

Choose 2 from 5.

Calculation: permutations are 5! / 3! = 120 / 6 = 20. Combinations divide by 2! to remove orderings: 20 / 2 = 10. So there are 10 ways to pick a pair and 20 ways to pick and arrange it. The same idea scales up: a 6-from-49 lottery has nearly 14 million combinations, which is why the odds of any single ticket are so long.

The choice between them comes down to whether order matters.

Ask whether reordering the same items counts as a new outcome. If yes, use permutations; if no, use combinations.

Every combination hides several permutations, one for each ordering.

Permutations equal combinations times r!, so as r grows the gap between them widens quickly.

The Size of N

More items to choose from raises both counts, often steeply.

How Many You Choose

The counts are largest when r is near half of n for combinations, and rise with r for permutations.

Whether Order Matters

Counting order multiplies the total by r!, so permutations far exceed combinations for larger r.

Probability

Count favorable and total outcomes for lottery, card or dice problems.

Planning and Design

Work out how many teams, committees or arrangements are possible.

Passwords and Codes

Count the possible orderings of digits or characters.

1. Using the Wrong One

If reordering makes a new outcome, use permutations; if not, combinations.

2. Choosing More than the Total

You cannot pick more items than you have, so r cannot exceed n.

3. Forgetting to Divide by R!

Combinations remove orderings by dividing the permutations by r!.

4. Allowing Repeats by Accident

These formulas assume no repeats. Choosing with replacement uses different formulas.

5. Rounding Large Counts

Big counts must stay exact. This tool uses big integers, so no precision is lost.

The counts are exact for whole numbers in range; only very long results are shortened for display.

A combination ignores order, so the same items in any order count once. A permutation counts order, so each arrangement is separate. Permutations are always at least as many as combinations.

Use n! divided by r! times (n minus r)!. It counts ways to choose r items from n when order does not matter. Choosing 2 from 5 gives 10 combinations.

Use n! divided by (n minus r)!. It counts ways to choose and arrange r items from n. Choosing 2 from 5 gives 20 permutations.

Use combinations when order does not matter, such as picking a team, a committee, or lottery numbers. Rearranging the same choice does not count as a new outcome.

Use permutations when order matters, such as ranking finishers, arranging letters, or setting a code. Each different order is a separate outcome.

Because each combination can be arranged in r! orders. Permutations equal combinations times r!, so the gap grows as r increases.

No. You cannot choose more items than are available, so r must be at most n. Choosing all n gives one combination.

No. These formulas assume each item is chosen at most once. Choosing with repetition, where an item can repeat, uses different formulas.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Combination (Wikipedia).
• Combinations and permutations (Maths Is Fun).
• Permutation (Wikipedia).