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Mathematics

Factorials: Why 0! = 1 and How n! Runs Counting

Permutations, the Empty Product, and Combinations Demystified — A TLDR Primer

Your teacher writes 0!=1 on the board and moves on like it's obvious. It isn't — until someone actually explains why. This TLDR primer walks through the factorial function from the ground up: what n! means, how the recursive definition works, and why the empty-product convention forces 0!=1 rather than making it an arbitrary rule to memorize.

From there it builds outward into everything factorials actually run: permutations (how many ways can you arrange a shelf of books), partial permutations P(n,k), and combinations — the C(n,k) formula behind Pascal's triangle and the binomial theorem. A dedicated section tackles the two things that trip students up on every combinatorics unit: arranging letters with repeats (MISSISSIPPI is the classic example) and circular arrangements, both handled cleanly with factorial ratios. The last stretch previews where n! shows up later — Stirling's approximation, Taylor series, probability, and the gamma function — so the concept doesn't feel like a dead end.

Written as a permutations and combinations guide for students who want the idea straight, not buried in a textbook chapter of unrelated proofs. Worked examples replace vague explanations, common misconceptions get named and corrected inline, and every term is defined the moment it appears. This is a combinatorics primer for high school and early college students facing a test on counting methods, plus parents or tutors who need to refresh the topic fast.

Short by design, stripped to essentials, and built to get you from confused to confident before your next quiz.

What you'll learn
  • Define n! and compute factorials by hand for small n
  • Explain why 0! = 1 using both the recursive definition and the empty-product convention
  • Use factorials to count permutations and arrangements with and without repetition
  • Derive and apply the combination formula C(n,k) = n!/(k!(n-k)!)
  • Recognize factorials inside the binomial theorem and Pascal's triangle
  • Estimate the growth rate of n! and know when to use Stirling's approximation
What's inside
  1. 1. What n! Means and How to Compute It
    Introduces the factorial as a product of descending integers, works small cases, and shows the recursive definition.
  2. 2. Why 0! = 1 (It's Not a Trick)
    Justifies the 0! = 1 convention through the recursion, the empty product, and the counting interpretation.
  3. 3. Factorials Run Counting: Permutations
    Shows how n! counts arrangements of n distinct objects and extends to partial permutations P(n,k).
  4. 4. Combinations and the Binomial Coefficient
    Derives C(n,k) from permutations by dividing out order, and connects it to Pascal's triangle and the binomial theorem.
  5. 5. Arrangements with Repeats and Restrictions
    Handles multiset permutations (like arranging MISSISSIPPI) and circular arrangements using factorial ratios.
  6. 6. How Fast n! Grows and Where It Shows Up
    Discusses the explosive growth of n!, introduces Stirling's approximation, and previews appearances in probability, Taylor series, and the gamma function.
Published by Solid State Press
Factorials: Why 0! = 1 and How n! Runs Counting cover
TLDR STUDY GUIDES

Factorials: Why 0! = 1 and How n! Runs Counting

Permutations, the Empty Product, and Combinations Demystified — A TLDR Primer
Solid State Press

Factorials: Why 0! = 1 and How n! Runs Counting

Permutations, the Empty Product, and Combinations Demystified — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What n! Means and How to Compute It
  2. 2 Why 0! = 1 (It's Not a Trick)
  3. 3 Factorials Run Counting: Permutations
  4. 4 Combinations and the Binomial Coefficient
  5. 5 Arrangements with Repeats and Restrictions
  6. 6 How Fast n! Grows and Where It Shows Up
Chapter 1

What n! Means and How to Compute It

The factorial of a positive whole number n, written n!, is the product of every whole number from n down to 1. In symbols:

n!=n×(n−1)×(n−2)×⋯×2×1

Read "n!" out loud as "n factorial." The exclamation point is just notation — it doesn't mean the number is exciting, though factorials do grow excitingly fast (more on that in a later section).

Let's compute a few by hand, because the pattern matters more than any single value:

1!=1 2!=2×1=2 3!=3×2×1=6 4!=4×3×2×1=24 5!=5×4×3×2×1=120

Notice you don't have to redo the whole multiplication each time. Since 4!=4×3×2×1 and 3!=3×2×1, you can see that 4!=4×3!. That's not a coincidence — it's the key structural fact about factorials, and it leads to a second, equally valid way to define them.

The recursive definition

Instead of writing out the whole product every time, you can define n! in terms of the factorial just below it. This is called a recursive definition — a definition that builds each value from the previous one, rather than spelling out the whole thing from scratch:

n!=n×(n−1)!

along with a starting point (a base case), 1!=1.

This says: "to get n!, take (n−1)! and multiply by n." Check it against the numbers above: 5!=5×4!=5×24=120. Matches. 4!=4×3!=4×6=24. Matches.

About This Book

If you're a high school student in Algebra II or Precalculus, a first-year in an intro statistics or discrete math course, or a parent helping with homework on counting problems, this book is for you. Anyone who has stared at n! on a page and needed factorial explained for students in plain language, without a professor's detour into abstract algebra, will feel at home here.

This is a combinatorics primer for high school and early college readers, covering how to calculate n factorial by hand, why is 0 factorial equal to 1 (the answer is logic, not convention), and how factorials drive a full permutations and combinations guide — arrangements, repeats, restrictions, and the binomial coefficient. You'll also get the binomial theorem explained simply and Stirling's approximation explained for when factorials get too large to compute directly. A concise overview with no filler.

Read it front to back, work through each worked example by hand, then test yourself on the problem set at the end before your exam or homework is due.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters — readable in one sitting.

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