Combination Calculator
Calculate combinations nCr — the number of ways to choose r items from n items where order does not matter.
nCr (Combinations)
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nPr (permutations)
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r!
r cannot be greater than n.
What is Combination Calculator?
A combination calculator computes nCr — the number of ways to choose r items from n items where order does not matter. Combinations are used in probability, statistics, and combinatorics. Unlike permutations, selecting items A, B, C and C, B, A count as the same combination because the arrangement is irrelevant.
How to use
- 1 Enter n, the total number of items available.
- 2 Enter r, the number of items to choose.
- 3 The main result shows nCr with the formula used.
- 4 The cards below display nPr (the related permutation count) and r! for reference.
- 5 r must not exceed n — an error message will appear if it does.
Formula
Example calculation
How many ways can a committee of 3 be chosen from 10 candidates? C(10, 3) = 10! / (3! x 7!) = 720 / 6 = 120. There are 120 possible committees. Note that C(10, 3) = P(10, 3) / 3! because combinations remove the ordering that permutations count.
Frequently asked questions
What is the difference between nCr and nPr?
nPr counts ordered arrangements, while nCr counts unordered selections. nCr = nPr / r!, because each combination of r items can be arranged in r! different orders, all of which are counted separately in permutations.
What does C(n, 0) equal?
C(n, 0) = 1 for any n. There is exactly one way to choose zero items from a set — by choosing nothing. This is consistent with 0! = 1 in the formula.
What does C(n, n) equal?
C(n, n) = 1. There is exactly one way to choose all n items from a set of n items.
How are combinations used in probability?
Combinations are used to count favorable outcomes in probability problems. For example, the probability of getting exactly 3 heads in 5 coin flips uses C(5, 3) = 10 favorable outcomes out of 2^5 = 32 total outcomes.
What is Pascals triangle and how does it relate?
Each entry in Pascals triangle is a combination value. The entry in row n and position r (0-indexed) equals C(n, r). This makes Pascals triangle a visual representation of all combination values and is useful in binomial expansions.