Probability Calculator – Combinations, Permutations & Factorials

A factorial n! multiplies every whole number up to n (10! = 3,628,800) and counts all orderings of n items. A permutation P(n, r) = n! ÷ (n − r)! counts ordered selections: P(10, 3) = 720 ways to pick and arrange a gold, silver and bronze medal from 10 runners. A combination C(n, r) = n! ÷ (r!(n − r)!) ignores order: C(49, 6) = 13,983,816 possible lottery tickets, which is why matching all six numbers is so unlikely. For probability, divide favourable outcomes by total outcomes: the chance of drawing 2 aces from a 52-card deck is C(4, 2) ÷ C(52, 2) = 6 ÷ 1326 ≈ 0.45%. Use combinations when order does not matter (committees, hands of cards) and permutations when it does (passwords, rankings); if repetition is allowed, use nʳ instead — 10 PIN digits give 10⁴ = 10,000 codes.

1. 1

   Choose a calculation type

   Pick combination, permutation or factorial.

2. 2

   Enter n and r

   For a lottery-style problem enter n = 49 total items and r = 6 selected.

3. 3

   Read the exact result

   C(49, 6) = 49! ÷ (6! · 43!) = 13,983,816 — every result is computed exactly, no rounding.

• 1

  Probability values always range from 0 (impossible) to 1 (certain)

• 2

  The sum of all probabilities in a sample space equals 1

• 3

  Use the multiplication rule for AND probabilities of independent events

• 4

  Use the addition rule for OR probabilities, subtracting overlap for non-mutually exclusive events

• 5

  Conditional probability P(A|B) = P(A and B) / P(B)

• Confusing independent and dependent events when multiplying probabilities

• Forgetting to subtract P(A and B) when using the addition rule for non-mutually exclusive events

• Assuming events are independent without verification

• Confusing probability with odds - they are different concepts

• Not considering all possible outcomes in the sample space