What is the difference between permutations and combinations?

In permutations, order matters (ABC ≠ BAC). In combinations, order does not matter (ABC = BAC). There are always fewer combinations than permutations.

How do I calculate P(5,3)?

P(5,3) = 5! / (5-3)! = 120 / 2 = 60. This counts ordered arrangements of 3 items from 5.

How do I calculate C(5,3)?

C(5,3) = 5! / (3! × 2!) = 120 / 12 = 10. This counts unordered selections of 3 items from 5.

How many ways can you choose 6 lottery balls from 40?

C(40,6) = 3,838,380. A classic application of combinations.

📐 Probability Calculator

Calculate permutations P(n,r) and combinations C(n,r) by entering n and r. The calculator shows the formula and result. In permutations order matters; in combinations it does not.

C(n,r) = n! / (r! × (n-r)!)

Unordered selections — order does not matter

Enter values

n (total number of items)r (number chosen)

C(10,3) =

120

This probability calculator works out permutations P(n,r) and combinations C(n,r) from the values you enter for n and r. It is a handy companion for math, statistics and combinatorics problems whenever you need to know how many ways elements can be selected or arranged.

How the calculator works and what it’s for▾

Permutation vs. combination

A permutation P(n,r) counts how many ways you can choose and arrange r items from a set of n when order matters. A combination C(n,r) counts the number of ways to choose r items when order does not matter.

Permutations use the formula P(n,r) = n! / (n−r)!, while combinations use C(n,r) = n! / (r! · (n−r)!). The exclamation mark denotes a factorial, for example 5! = 5·4·3·2·1 = 120.

What you enter and what you get

Enter two whole numbers: n is the total number of items in the set and r is how many you select, where r must be no greater than n.

The calculator returns both the permutation and the combination count, often shown alongside the formula. You can instantly see how many distinct arrangements or selections exist for your scenario.

A worked example

Suppose a group has 5 people (n = 5) and you pick 3 of them (r = 3). If order matters, such as awarding gold, silver and bronze medals, there are P(5,3) = 60 possibilities.

If order does not matter, such as forming a team of three, there are only C(5,3) = 10 possibilities. The same starting point gives very different answers depending on whether order counts.

Where the calculator helps

Permutations and combinations appear in classical probability, lottery and raffle odds, estimating the number of possible passwords or codes, and many problems in statistics and discrete mathematics.

A common mistake is confusing the two. Always ask first whether order affects the outcome. If it does, use a permutation; if it does not, use a combination.

🔄 Reviewed June 2026