Combination Calculator
Calculate combinations (nCr) easily online for probability, statistics, and mathematics
Combination Calculator
Please enter a valid positive integer
Please enter a valid positive integer less than or equal to n
Understanding Combinations
Combinations are a fundamental concept in mathematics, particularly in probability and statistics. They represent the number of ways to select items from a larger set where the order of selection does not matter.
Combinations Formula
The formula for calculating combinations is:
C(n, r) = n! / (r! × (n - r)!)
Where:
- n is the total number of items
- r is the number of items to choose
- ! represents the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)
Example Calculation
Problem: How many ways can you choose 3 books from a shelf of 10 books?
Solution: Using the combinations formula:
C(10, 3) = 10! / (3! × (10-3)!) = 10! / (3! × 7!) = 120
There are 120 different ways to choose 3 books from 10.
When to Use Combinations
Use combinations when:
- The order of selection doesn't matter
- You're selecting a subset from a larger set
- Each item can be selected only once
Common applications include lottery probabilities, committee selections, and card game probabilities.
Frequently Asked Questions
What's the difference between combinations and permutations?
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Combinations consider the selection of items without regard to order, while permutations consider the arrangement of items where order matters. For example, selecting 3 people for a committee is a combination, but arranging 3 people in specific roles is a permutation.
Can r be greater than n in combinations?
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No, in standard combinations, r cannot be greater than n. If you try to select more items than are available, the result is 0. Our calculator will show an error if r > n.
What is the value of C(n, 0) and C(n, n)?
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C(n, 0) = 1 (there's exactly one way to choose nothing)
C(n, n) = 1 (there's exactly one way to choose all items)
C(n, n) = 1 (there's exactly one way to choose all items)