Combination Calculator (nCr)

Use combinations when only the selection matters, not the order: committees, teams, pizza toppings, sets of questions.

Combinations

Combinations C(n, r)13,983,816
Permutations P(n, r) for comparison
10,068,347,520

Working

  1. C(n, r) = n! ÷ (r! × (n − r)!) = 49! ÷ (6! × 43!)

Formulas

without repetition: nCr = n! ÷ (r! × (n − r)!)

with repetition: (n + r − 1)! ÷ (r! × (n − 1)!)

Worked example

Choosing a 6-person committee from 49 members

  1. n = 49, r = 6, no repetition
  2. 49! ÷ (6! × 43!) = (49 × 48 × 47 × 46 × 45 × 44) ÷ 720

13,983,816 possible committees — so any one particular group of six has a 1 in 13,983,816 chance of being picked at random.

3 scoops from 5 flavors, repeats allowed

  1. n = 5, r = 3, with repetition
  2. (5 + 3 − 1)! ÷ (3! × 4!) = 7! ÷ (6 × 24)

35 different bowls.

Symmetry

Choosing r items to take is the same as choosing n − r items to leave behind, so nCr = nC(n − r). For example 10C3 = 10C7 = 120. The values of nCr form Pascal’s triangle.