Combination Calculator

Calculate the number of possible selections C(n,r) with step-by-step solutions

Input Values

Result

What are Combinations?

A combination is a selection of objects from a set where order does not matter. The number of combinations of n objects taken r at a time, denoted as C(n,r), represents the number of different ways to select r objects from a set of n distinct objects without considering the order.

Combination Formula

The formula for calculating combinations is:

C(n,r) = n! / (r! × (n-r)!)

Where:

  • n = Total number of objects
  • r = Number of objects to select
  • n! = n factorial (n × (n-1) × (n-2) × ... × 1)
  • r! = r factorial
  • (n-r)! = (n-r) factorial

When to Use Combinations

  • When order doesn't matter (e.g., selecting committee members)
  • When choosing items from a group
  • For lottery number selections
  • For team selections
  • For sampling without replacement

How to Use the Calculator

  1. Enter the total number of objects (n)
  2. Enter the number of objects to select (r)
  3. Click "Calculate" to get the number of combinations
  4. View the step-by-step solution

Example Calculations

C(5,3):

n = 5, r = 3

C(5,3) = 5! / (3! × 2!)

C(5,3) = 120 / (6 × 2) = 10

C(6,2):

n = 6, r = 2

C(6,2) = 6! / (2! × 4!)

C(6,2) = 720 / (2 × 24) = 15

Important Notes

  • r must be less than or equal to n
  • 0! = 1 (by definition)
  • C(n,n) = 1 (all objects selected)
  • C(n,1) = n (one object selected)
  • C(n,0) = 1 (no objects selected)
  • C(n,r) = C(n,n-r) (symmetry property)

Difference Between Combinations and Permutations

Combinations (C(n,r)):

Order doesn't matter

ABC = BAC

Formula: n! / (r! × (n-r)!)

Permutations (P(n,r)):

Order matters

ABC ≠ BAC

Formula: n! / (n-r)!