Calculate the number of ordered arrangements of r items chosen from n items using the nPr formula.
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Permutations (nPr)
There are 720 ordered arrangements when choosing 3 items from 10.
Permutations vs. combinations
Permutations count ordered arrangements — the sequence matters (e.g., ABC is different from BAC). Combinations count unordered selections — only the chosen items matter, not their order. P(n,r) is always greater than or equal to C(n,r) because each combination corresponds to r! permutations.
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Combinatorics
A permutation calculator finds the number of ways to arrange r items chosen from a set of n items when the order matters. It uses the nPr formula, also written as P(n,r), which multiplies consecutive descending integers from n down for r terms.
A permutation counts arrangements where order is significant. Arranging A, B, C is different from C, A, B. The formula multiplies n × (n−1) × (n−2) × … for r factors, which equals n! / (n−r)!.
For example, P(10,3) = 10 × 9 × 8 = 720. There are 720 different ways to arrange 3 items chosen from 10.
P(n, r) = n! / (n − r)!
The number of ordered arrangements of r items chosen from n items.
Use permutations when the order of selection matters: assigning 1st, 2nd, and 3rd place in a race, creating PIN codes, arranging books on a shelf, or sequencing tasks. The key question is whether rearranging the same items produces a different result.
For a 4-digit PIN using digits 0–9 without repetition, the number of possible PINs is P(10,4) = 10 × 9 × 8 × 7 = 5,040. If repetition is allowed, the count is 10^4 = 10,000.
Frequently asked questions
Permutations count ordered arrangements where sequence matters (ABC ≠ BAC). Combinations count unordered selections where only membership matters (ABC = BAC). P(n,r) is always greater than or equal to C(n,r) because each combination maps to r! permutations.
P(n, 0) = 1 for any non-negative n. There is exactly one way to arrange zero items: the empty arrangement.
The standard nPr formula assumes all items are distinct. For arrangements with repeated items, divide by the factorial of each group of identical items. For example, the number of arrangements of the letters in MISSISSIPPI is 11! / (4! × 4! × 2!).
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