Calculate permutations (nPr), combinations (nCr), and factorials instantly.
This tool calculates the number of permutations (ordered arrangements) or combinations (unordered selections) possible from a set of items. It's fundamental in probability, statistics, and combinatorics problems.
Permutations: P(n,r) = n! ÷ (n−r)!, where order matters. Combinations: C(n,r) = n! ÷ (r!(n−r)!), where order doesn't matter. Here, n is the total number of items and r is the number chosen.
Enter the total number of items (n) and how many you're selecting or arranging (r), then choose whether order matters (permutation) or not (combination), and the calculator returns the result.
Example: Choosing 3 winners from 10 people where order doesn't matter: C(10,3) = 10!/(3!×7!) = 120 combinations. If order does matter (1st, 2nd, 3rd place): P(10,3) = 10!/7! = 720 permutations.
What's the key difference between permutations and combinations? Permutations count arrangements where order matters (like race finishing positions), while combinations count selections where order doesn't matter (like choosing a committee) — this is why permutations always give equal or larger results than combinations for the same n and r.
Why is factorial (!) used in these formulas? Factorial represents the total number of ways to arrange all n items in sequence, which forms the foundation for calculating more specific arrangements or selections of a subset.
When would I use combinations instead of permutations? Use combinations when the order of selection doesn't matter, such as choosing lottery numbers, forming a team, or selecting a subset of items where their arrangement isn't relevant to the outcome.
What real-world problems use permutations and combinations? These concepts appear in probability calculations, password/PIN possibilities, scheduling problems, genetics (gene combinations), and card game probability calculations.
Can r be larger than n? No, both permutations and combinations require r ≤ n, since you cannot select or arrange more items than actually exist in the original set.