Permutation Without Repetition Calculator

This calculator counts ordered selections made from a set of distinct objects when an object cannot be chosen twice. It answers questions such as how many three-place finishing orders are possible among eight runners, or how many different four-character arrangements can be made from ten distinct symbols. Because order matters, ABC and BAC count as different outcomes.

The phrase “without repetition” describes the selection rule. Once an object has been used in one position, it is unavailable for later positions. That makes this calculation different from a power such as n^r, which allows the same object to appear again, and from a combination, which ignores order.

1. Enter the total number of distinct available objects as n.
2. Enter the number of ordered positions to fill as r. Both entries must be whole numbers from zero through 100, and r cannot exceed n.
3. Select Calculate. The result is formatted as a whole-number count.

Before entering values, decide what makes two outcomes different. If swapping the first and second selected objects creates a new outcome, a permutation is appropriate. If that swap still represents the same group, use a combination instead.

The page multiplies r descending factors beginning at n. The factorial form is a compact way to write that product. A selection of zero objects has one arrangement, the empty arrangement; this convention keeps many counting identities consistent.

The output is a count of possible sequences, not the probability of any one sequence. If all 336 medal orders in the example were equally likely, one specified order would have probability 1/336, but equal likelihood is an additional assumption rather than part of the counting formula.

Counts grow quickly. A large answer may be mathematically correct even when n looks modest. Use the result to describe the size of the outcome space, then apply any separate restrictions—such as prohibited positions or tied participants—outside this basic calculation.

For a practical check, compare neighboring sample sizes. Moving from r positions to r + 1 positions multiplies the count by n − r, the number of objects still available. This gives a quick way to spot a missing factor.