Combination Calculator Formula

Understand the math behind the combination calculator. Each variable explained with a worked example.

Use the Combination Calculator

Formulas Used

C(n, r)

combinations = factorial(n) / (factorial(r) * factorial(n - r))

P(n, r) for comparison

permutations = factorial(n) / factorial(n - r)

Variables

VariableDescriptionDefault
nTotal Items (n)10
rItems Chosen (r)3

How It Works

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.

Worked Example

How many ways can you choose a committee of 3 from 10 people?

n = 10r = 3
  1. 01C(10, 3) = 10! / (3! * 7!)
  2. 02= (10 * 9 * 8) / (3 * 2 * 1)
  3. 03= 720 / 6 = 120
  4. 04Compare with permutations: P(10,3) = 720

Frequently Asked Questions

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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