How to Calculate Permutations and Combinations (2026)

Permutations and combinations both count ways to select items from a set, but they answer different questions. A permutation counts arrangements where order matters — the sequence ABC is different from BAC. A combination counts selections where order does not matter — picking {A, B, C} is the same as picking {C, B, A}. Getting this distinction right is the first step in any counting problem. Use the Permutation & Combination Calculator to compute results instantly without manual arithmetic.

The Formulas

Both formulas are built from factorials. The factorial of n (written n!) is the product of all positive integers up to n — so 5! = 5 × 4 × 3 × 2 × 1 = 120.

• Permutations (nPr): P(n, r) = n! / (n − r)! — how many ways to arrange r items chosen from n distinct items.
• Combinations (nCr): C(n, r) = n! / (r! × (n − r)!) — how many ways to choose r items from n without caring about order.

The only difference between the two formulas is the extra r! in the denominator of combinations. That factor divides out all the duplicate arrangements that combinations treat as identical.

Worked Examples

Consider a set of 5 items (n = 5) and selecting 3 of them (r = 3):

There are 60 ordered arrangements but only 10 unique groups — exactly 3! = 6 times fewer, because each group of 3 can be arranged in 6 ways.

When to Use Each

Use a permutation when the sequence matters: passcodes, race finishing positions, assigning distinct roles to people. Use a combination when only the selection matters: lottery draws, choosing a committee, picking toppings on a pizza. A quick test — ask “does swapping two items give a different outcome?” If yes, use permutations; if no, use combinations.

Use the Permutation & Combination Calculator to handle any values of n and r instantly, including large numbers where factorials become unwieldy to compute by hand.