Combinations Permutations Calculator

The combinations and permutations calculator computes nCr and nPr for any non-negative integers n and r where r ≤ n. It uses arbitrary-precision arithmetic for exact integer results and shows step-by-step factorial cancellation.

Combinations vs Permutations

Combinations (nCr) count unordered selections. For example, choosing a 3-person committee from 10 people — the order of selection does not matter. The formula is:

C(n, r) = n! / (r! × (n - r)!)

Permutations (nPr) count ordered arrangements. Awarding gold, silver, and bronze medals to 3 of 10 finalists — the order matters. The formula is:

P(n, r) = n! / (n - r)!

How to Use

Select either combinations or permutations. Enter the total number of items (n) and the number of items to choose (r). The calculator displays the result with a step-by-step breakdown showing the factorial cancellation process. Results update in real time.

Key Properties

• P(n, r) = C(n, r) × r! — each unordered selection can be arranged in r! distinct orders.
• C(n, 0) = C(n, n) = 1 — exactly one way to choose nothing or everything.
• C(n, r) = C(n, n - r) — symmetry property, choosing r items is the same as leaving n - r items out.
• The calculator supports n up to 170, after which factorial values exceed JavaScript's safe integer range.