Combinations Calculator
Calculate the number of combinations C(n, r) — choosing r items from n without regard to order. See also Permutations Calculator and Factorial Calculator.
How to Calculate Combinations
A combination is a selection of items where order does not matter. To calculate C(n, r), divide n! by the product of r! and (n−r)!. For example, choosing 3 people from a group of 10 — it does not matter in what order they are chosen, only which 3 are selected. This is different from permutations, where order matters.
Combinations Formula
C(n, r) = n! / (r! × (n − r)!)
where:
n = total number of items
r = number of items chosen
n! = n factorial = n × (n-1) × ... × 1
Example
C(10, 3) = 10! / (3! × 7!)
= 3628800 / (6 × 5040)
= 3628800 / 30240
= 120
There are 120 ways to choose 3 items from 10.
Frequently Asked Questions
What is the difference between combinations and permutations?
Combinations count selections where order does not matter (e.g., choosing a committee). Permutations count arrangements where order matters (e.g., assigning 1st, 2nd, 3rd place). P(n,r) = C(n,r) × r!.
What is C(n, 0) and C(n, n)?
Both equal 1. There is exactly one way to choose nothing (empty set) and exactly one way to choose everything (the full set).
What is Pascal's Triangle?
Pascal's Triangle is a triangular array where each entry is a combination value C(n, r). Each number is the sum of the two numbers directly above it. Row n contains C(n, 0) through C(n, n).
Where are combinations used in real life?
Combinations are used in lottery probability, card games (poker hands), genetics (gene combinations), statistics (sampling), and computer science (algorithm analysis).