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Official Data Lab

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STATLABAcademyStatistics through data and R practice

All STATLAB Academy lessons, quizzes, datasets, and notes are original or properly licensed. Textbooks may be used only as curriculum references.

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STATLAB Academy | Learn Statistics with R and Real-World Data

Lesson 15 of 16

The Fundamental Counting Rule and Factorials

Home/Modules/Module 5/The Fundamental Counting Rule and Factorials

Count stage-by-stage configurations and ordered arrangements with validated nonnegative integers.

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40 minutes94% path

Learning objectivesBy the end of this lesson, students will be able to:

1Define the events and probability quantities used in the fundamental counting rule and factorials.
2Select and calculate an appropriate probability rule with a visible denominator or assumption.
3Interpret the result for a business decision without overstating certainty or causation.

Theory

Core probability reasoning

Count stage-by-stage configurations and ordered arrangements with validated nonnegative integers.

Explanation 1

Product rule: Multiply choices available at successive stages.

Explanation 2

Factorial: The number of orders for n distinct items.

Explanation 3

Zero factorial: The standard empty-arrangement count.

Terminology

Concept map

Product rule

Multiply choices available at successive stages.

Region x channel x priority.

Factorial

The number of orders for n distinct items.

n!.

Zero factorial

The standard empty-arrangement count.

0! = 1.

Notation and formulas

Counting rules

Key relationship

total = n1 x n2 x ... x nk; n! = n(n - 1)...1

Define every event before substituting values, then validate the result and denominator.

Step 1List stages and allowed choices.
Step 2State whether repetition is allowed.
Step 3Multiply or use factorial as appropriate.

Interactive Mission

counting rule builder

Stage 1 choicesStage 2 choicesStage 3 choices

Product-rule configurations

24

Arrangements of stage 1 items

6

0! = 1

State whether repetition is allowed and whether each stage has the same available choices.

Worked example

Campaign configuration

Scenario

A campaign chooses a segment, message, and delivery window.

A large count does not assign equal probabilities.

R connection

R note: Counting Rules

Run and edit the code, inspect intermediate output, and use the movable R Code Guide while practising.

Live R Lab

Counting Rules

Compare product rules, factorials, permutations, and combinations with validated inputs.

Editable R code

Results

Ready to run

Ready to run

Ready to run. Results, errors, tables, plots, and fallback messages will appear here.

Validate n and r.

Apply the product rule.

Distinguish order-sensitive and order-insensitive selections.

Dataset provenance

Original synthetic data generated for STATLAB Academy. No textbook data used.

Interpretation safeguard

Clearly identify whether repetition is allowed and whether every position has the same choices.

Reflection

Pause and write

Where could an unclear event definition or assumption change the conclusion in the fundamental counting rule and factorials?

Name the event, denominator, probability interpretation, and one operational limitation.

Assistant

Ask your probability coach

Help me reason through The Fundamental Counting Rule and Factorials by asking questions about my event definitions, denominator, formula choice, assumptions, and interpretation. Do not give a quiz answer before I submit.

Exit check

Lesson checkpoint

Use a hint before submitting if needed. Full calculations and interpretations appear after submission.

Checkpoint

Module 5 Lesson 15 Checkpoint

Question 1 of 2. Answered 0/2. Passing score: 70%.

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Question 1: Numeric answer

mediumCalculate and responsibly interpret probability for a defined event.

A service code has 3 region choices, 4 channel choices, and 2 priority choices. How many codes are possible?

Resource

Recommended reading

Explore additional probability explanations and exercises from properly licensed or publisher resources.

Resource

Module 5 learning material

Download an original worksheet or guide for this lesson.

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