Combinatorics Calculator

Compute permutations, combinations, multichoose, and derangements with formula outputs.

Scratchpad (not saved)

Save to create a persistent, shareable workspace.

What This Calculator Does

This calculator packages the most common counting formulas into one place so you can switch between arrangements, selections with or without order, repeated selections, and derangements. It is useful for both classroom combinatorics and practical counting problems.

It combines n, r, Operation to estimate Operation, Result, Formula.

Formula & Method

Core equations: Permutations

P(n,r)=\frac{n!}{(n-r)!}

; combinations

C(n,r)=\frac{n!}{r!(n-r)!}

; multichoose

\left(\!\!\binom{n}{r}\!\!\right)=\binom{n+r-1}{r}

; derangements

!n=n!\sum_{k=0}^{n}\frac{(-1)^k}{k!}

Notation used in the formulas: $R$ = Operation; $x_{1}$ = n; $x_{2}$ = r; $x_{3}$ = Operation.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it whenever you need to decide whether order matters, repetition is allowed, or no item is allowed to remain fixed, and want the correct counting formula without re-deriving it by hand.

Reference Book

Introduction to Probability

Second Edition

Joseph K. Blitzstein and Jessica Hwang · Chapman and Hall/CRC

A widely used probability text for counting, conditioning, expected value, core distributions, and simulation.

View Book

Inputs Used

n: Used directly in the calculation.
r: Used directly in the calculation.
Operation: Used directly in the calculation.

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