Combination Calculator
Compute combinations (nCr) and permutations (nPr) with exact integer results. Free online combination and permutation calculator.
Calculation Type
n (total items)
n
r (choose/arrange)
r
Result (nCr)
120
10C3 = 10! / (3! × 7!) = 120
How It Works
A combination (nCr) counts the number of ways to choose r items from a set of n without regard to order: nCr = n! / (r!(n − r)!). A permutation (nPr) counts the number of ways to arrange r items chosen from n, where order matters: nPr = n! / (n − r)!.
This tool uses an iterative factorial to avoid overflowing for larger values of n, giving exact integer results.
Example
Choosing 3 books out of 5 to take on a trip yields 5C3 = 10 possible combinations, but arranging 3 of those 5 books on a shelf yields 5P3 = 60 permutations.
Common Use Cases
- Counting possible selections for teams, committees, or prize winners.
- Probability problems in statistics and gaming.
- Arranging subsets where order matters (passwords, racing podiums).
FAQs
What is the difference between nCr and nPr?
nCr (combinations) ignores order, while nPr (permutations) counts arrangements where order matters. For the same n and r, nPr is always larger or equal to nCr.
What does n choose r mean?
"n choose r" (nCr) is the number of distinct r-item subsets you can form from a set of n items, regardless of the order you pick them.
Why use an iterative factorial?
Iterative multiplication avoids the recursion stack and produces exact integers, which is preferable for very large n where floating-point precision would otherwise be lost.