CALCULATE WITH CONFIDENCE

Counting, Permutations & Combinations

Fundamental Counting Principle Calculator

Multiply independent stage counts to find the total number of possible outcomes.

Counting & Combinatorics

Enter the number of choices at each stage

Product rule
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example
Enter one non-negative count per line or separate counts with semicolons.

Total possible outcomes

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Enter valid values to see the result.

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Understand the subject

What is the fundamental counting principle?

The product rule counts complete outcomes built from independent stages. Every choice at one stage can be paired with every choice at the next.

Use it for outfits, codes, menus, schedules, and other staged choices.

The relationship

The counting rule

See the structure

Every branch continues through every later stage

Worked example

Check the rule with a small case

3 shirts, 4 trousers, and 2 shoe choices make 3 × 4 × 2 = 24 outfits.

Interpret with care

Choose the right counting model

Only multiply when each stage can genuinely follow every earlier choice. Restrictions remove branches.

Use the Permutation & Combination Generator when a small case needs to be inspected rather than only counted.

Quick guide

How to use this calculator

  1. Enter the number of choices available at each stage.
  2. Use one row for each independent stage.
  3. Read the exact product of all stage counts.

Calculation method

Apply the product rule

When every choice at one stage can be followed by every choice at the next stage, the total number of outcomes is the product of the stage counts.

Worked example

Build an outfit

Three shirts, four trousers, and two pairs of shoes produce 24 outfits.

3×4×2 = 24

Supported inputs

Precision and limits

Discrete domain

Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.

Exact integers

Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.

Stages

Up to 100 non-negative stage counts and 10,000 entered digits are supported.