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Work out chances, permutations, and combinations — with every count shown.
Count the favourable outcomes, then divide by all equally likely outcomes.
Choose Simple probability, Permutations, or Combinations and enter whole-number counts. Press Calculate to see the working; for counting modes use 0 ≤ r ≤ n ≤ 12, and for simple probability make the total positive and the favourable count no greater than it.
With equally likely outcomes, probability = favourable outcomes ÷ total outcomes. Permutations count ordered selections without replacement: nPr = n!/(n − r)!. Combinations count unordered selections without replacement: nCr = n!/[r!(n − r)!]. A factorial multiplies a positive integer by every smaller positive integer, with 0! defined as 1.
A fair six-sided die has three even outcomes: 2, 4, 6. Enter favourable 3 and total 6: P(even) = 3/6 = 1/2 = 50%.
Choose Permutations with n = 5 and r = 2. There are 5 choices for first place and 4 for second, so 5P2 = 5 × 4 = 20.
Choose Combinations with n = 5 and r = 2. Each unordered pair appears twice in the ordered count, so 5C2 = 20/2 = 10.
Use permutations when changing the order changes the outcome, such as first and second places. Use combinations for a group where the same members count once regardless of order.
Not with these counting modes. They assume distinct items selected without replacement; arrangements such as repeated digits need a different formula.
Counting alone gives a probability only when each counted outcome has the same chance. A biased die or unequal item weights cannot be modeled merely by favourable and total counts.
The current counting implementation deliberately caps n at 12 for exact small counts. The limit applies to permutations and combinations, not to the same-size total in simple probability. Choosing zero items gives one empty selection.
Convert chance fractions with the Fraction and Percentage calculators. The probability lesson develops sample spaces and equally likely outcomes. Related tools: Fraction Calculator, Percentage Calculator. Read the Probability lesson or practice probability to build confidence without relying on a calculator.