Combination Calculator (nCr)

About Combinations

In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter.

Combination Formula

C(n, r) = n! / [r! × (n – r)!]

Where:

  • n = total number of objects in the set
  • r = number of objects selected from the set
  • ! = factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)

Understanding Combinations

Combinations differ from permutations in that the order of selection does not matter. For example, selecting fruits for a fruit salad is a combination – choosing apples, bananas, and oranges is the same as choosing bananas, oranges, and apples.

Example Calculation

If you have 5 different books and want to choose 2 to take on a trip, the number of combinations is:

C(5, 2) = 5! / [2! × (5-2)!] = 120 / [2 × 6] = 10

So there are 10 different ways to choose 2 books from 5.

Applications of Combinations

Combinations are used in various fields including:

  • Probability and statistics
  • Lottery and gambling calculations
  • Computer science algorithms
  • Genetics and biological studies
  • Business decision making