Permutation (nPr) considers the order of items as important — different arrangements count as different outcomes. For example, ABC and CBA are distinct permutations.

Combination (nCr) ignores order — only the selection of items matters. ABC and CBA are the same combination. Use permutations when arranging items in sequence (like race rankings); use combinations when simply selecting a group (like lottery numbers).

The permutation formula is:
P(n, r) = n! / (n − r)!

This equals n × (n−1) × (n−2) × ... × (n−r+1) (exactly r factors). It counts the number of ways to arrange r items selected from a set of n distinct items, where order matters.

The combination formula is:
C(n, r) = n! / (r! × (n − r)!)

This can also be written as C(n, r) = P(n, r) / r!. It gives the number of ways to choose r items from n items where the order does not matter. It's also called the binomial coefficient and is often written as ⁿCᵣ or C(n, r).

Use permutations (nPr) when order matters:

• Ranking winners in a race (1st, 2nd, 3rd)
• Creating PIN codes or passwords
• Arranging books on a shelf
• Assigning specific seats to people

Use combinations (nCr) when order doesn't matter:

• Selecting lottery numbers
• Choosing committee members
• Picking toppings for a pizza
• Forming a hand in card games

No. For both nPr and nCr, you must have n ≥ r ≥ 0. If r > n, the calculation is mathematically undefined because you cannot select more items than are available. Our calculator will show a validation error if you try to enter r > n.

0! = 1 by convention. This definition ensures that the permutation and combination formulas work correctly for boundary cases. For example, nP0 = 1 (there is exactly one way to arrange zero items) and nC0 = 1 (there is exactly one way to choose zero items). This also makes nCn = n!/(n!×0!) = 1, which is consistent.

The relationship is: nPr = nCr × r!

In other words, a permutation is a combination followed by arranging the selected items. First choose r items from n (that's nCr), then arrange those r items in all possible orders (r! ways). This gives the total number of permutations: nCr × r! = nPr.

This calculator supports values of n up to 1,000 and r from 0 to n. It uses arbitrary-precision integer arithmetic (BigInt) to deliver exact results even when the numbers are extremely large — far beyond what standard floating-point calculators can handle accurately. Results may contain hundreds or even thousands of digits for large inputs.

Permutations: Password strength analysis, tournament bracket arrangements, DNA sequence ordering, scheduling tasks, cryptography key generation.

Combinations: Lottery probability calculations, quality control sampling, portfolio diversification in finance, clinical trial group assignment, poker hand probability, and any scenario involving selecting subsets without regard to order.