Combinations and permutations calculator (nCr, nPr, n!)
C(n, k), P(n, k) and n! as exact values, computed in integers, with the rule that separates combinations (order ignored) from permutations (order counted).
Result
2,598,960
- Formula
- C(n, k) = n! / (k! × (n − k)!)
- Arrangements P(52, 5)
- 311,875,200
- Ordering factor 5!
- 120
- Number of digits
- 7
- Order of magnitude
- 2.59896 × 10⁶
Note
5 objects chosen out of 52, unordered.
Calculation, assumptions and sources
OverviewCombinations C(n, k) (nCr) when order does not matter, permutations P(n, k) (nPr) when it does, and the factorial n!, as exact values: 100!
Combinations C(n, k) (nCr) when order does not matter, permutations P(n, k) (nPr) when it does, and the factorial n!, as exact values: 100! has 158 digits.
Method & formulasC(n, k) = n! / (k! × (n − k)!)
With 10 runners, a podium is a permutation, P(10, 3) = 720; a qualified trio is a combination, C(10, 3) = 120. One trio makes 3! = 6 podiums. Choosing k objects amounts to discarding n − k: C(n, k) = C(n, n − k).
- Combination, “n choose k”
- an unordered subset of k objects taken from n. C(n, k) is the binomial coefficient of the expansion of (a + b)ⁿ and of the binomial distribution.
- Permutation
- an ordered sequence of k distinct objects taken from n, without repetition.
- 0! = 1
- convention: there is one way to order zero objects, hence C(n, 0) = C(n, n) = 1.
Validity domainDistinct objects, without repetition.
Distinct objects, without repetition. Arrangements with replacement (nᵏ), combinations with repetition and permutations of partially identical objects (anagrams) are not covered. n is capped at 5,000.