Combinations & Permutations Calculator

This calculator counts the number of ways to choose or arrange items — the core of combinatorics and probability. It handles three operations: combinations (nCr) where order does not matter, permutations (nPr) where it does, and plain factorials (n!). It is useful for lottery odds, password-strength sums, statistics homework and any “how many ways” question.

How it works

The three modes use standard combinatorial formulas:

• Factorial: n! = n × (n−1) × … × 2 × 1, the number of orderings of n items.
• Permutations: nPr = n! ÷ (n − r)!, computed iteratively as n × (n−1) × … × (n−r+1) to avoid overflow.
• Combinations: nCr = n! ÷ [r! × (n − r)!], computed step by step (using the smaller of r and n−r) so larger inputs stay accurate.

Because order matters in permutations but not in combinations, nPr is always at least as large as nCr for the same n and r.

Example

Choosing 3 items from 10:

nCr(10, 3) = 10! ÷ (3! × 7!) = 120 nPr(10, 3) = 10 × 9 × 8 = 720

So there are 120 unordered selections but 720 ordered arrangements — the permutation count is 3! = 6 times larger, because each combination can be ordered in 6 ways.

Choose a mode, enter your values for n and r, and the result appears instantly — all computed locally in your browser with nothing uploaded.

Real-world applications

Lottery odds — a 6-from-49 lottery uses combinations, since the order the balls come out doesn’t matter. nCr(49, 6) gives the total number of possible tickets: 13,983,816. That is your denominator for calculating the probability of a jackpot win.

Password and PIN strength — a 4-digit PIN drawn from 10 digits (0–9) with no repeats uses permutations: nPr(10, 4) = 5,040 possible PINs. If repeats are allowed, it is simply 10^4 = 10,000. The permutation count is lower because eliminating repeats reduces the pool.

Team selection — choosing 5 players from a squad of 15 for a starting lineup where roles are not assigned uses combinations: nCr(15, 5) = 3,003 ways. If you are assigning 5 players to 5 specific positions (goalkeeper, defender, etc.), you need permutations: nPr(15, 5) = 360,360.

Card hands — a 5-card poker hand from a 52-card deck: nCr(52, 5) = 2,598,960 possible hands. Order doesn’t matter in a dealt hand, so combinations apply.

A/B testing variants — if you have 8 design elements and want to test all pairs simultaneously, nCr(8, 2) = 28 pairs. Knowing this number upfront prevents experimental designs that grow unmanageable.

Understanding the relationship between nCr and nPr

For any n and r, nPr = nCr × r!. This makes intuitive sense: for every unordered combination of r items, you can arrange those same items in r! different orders, each arrangement being a distinct permutation. For example, choosing 3 from 10: nCr(10, 3) = 120 and nPr(10, 3) = 720 = 120 × 3! = 120 × 6.

Edge cases to know

• nCr(n, 0) = 1 for any n — there is exactly one way to choose nothing.
• nCr(n, n) = 1 — there is exactly one way to choose all items.
• nCr(n, 1) = n — choosing a single item from n has n possibilities.
• Factorials grow extremely fast: 20! is about 2.4 × 10^18. The calculator handles moderately large inputs using iterative multiplication to avoid overflow, but very large factorials (above ~170) exceed JavaScript’s number range.