Combination Calculator
Calculate the number of combinations (unordered selections) when choosing r items from n items.
C(n, r)
120
C(n, r) vs Total Items (n)
Formula
## How to Calculate Combinations ### Formula **C(n, r) = n! / (r! * (n - r)!)** A combination counts the number of ways to choose r items from n distinct items where order does not matter. For example, choosing 3 team members from 10 candidates is a combination problem because the group {A, B, C} is the same regardless of selection order.
Esempio Risolto
How many ways can you choose a committee of 3 from 10 people?
- 01C(10, 3) = 10! / (3! * 7!)
- 02= (10 * 9 * 8) / (3 * 2 * 1)
- 03= 720 / 6 = 120
- 04Compare with permutations: P(10,3) = 720
Domande Frequenti
Why is C(n,r) always less than or equal to P(n,r)?
Because C(n,r) = P(n,r) / r!. Each unordered combination corresponds to r! ordered permutations, so dividing removes the duplicate orderings.
What is C(n, 0) and C(n, n)?
Both equal 1. There is exactly one way to choose nothing (the empty set) and exactly one way to choose everything.
What is the relationship to Pascal's triangle?
The entry in row n, position r of Pascal's triangle equals C(n, r). The recursive identity is C(n, r) = C(n-1, r-1) + C(n-1, r).
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