What This Calculator Does

This probability calculator covers four common but non-interchangeable tasks: permutation counts, combination counts, exact-k binomial probabilities, and exact-k Poisson probabilities. The selector puts “how many arrangements or selections?” and “how likely is an exact event count?” in one place without pretending they are the same calculation.

Choose the matching mode when you need nPr, nCr, binomial P(X=k), or Poisson P(X=k). The binomial and Poisson modes also show expectation, variance, standard deviation, and coefficient of variation so you can compare a distribution’s center and relative spread. For P values, use the P Value Calculator; for standardized scores, use the Z-Score Calculator; for ordinary numeric expressions, use the Mathematical Expression Calculator.

When to Use It

• You need a permutation or combination count and know whether order matters.
• You have a fixed number of independent binary trials with success probability p and need exactly k successes.
• You model event counts with an average rate λ per interval and need exactly k events.
• You want expectation, variance, standard deviation, and relative variability alongside the headline result.
• You have already selected the model and need a quick, checkable numerical result.

Do not treat this page as a general probability solver for conditional probability, Bayes’ theorem, event unions/intersections, normal, geometric, or hypergeometric distributions, or at-least/at-most cumulative probabilities.

Inputs Explained

Permutation and Combination: n and r

Both modes require non-negative integer n and r, with r ≤ n.

• Permutation nPr: different orders count as different outcomes, such as assigning first, second, and third place.
• Combination nCr: only the selected members matter, such as choosing a committee.