LaTeX
\max_i n_i\ge\left\lceil\frac{N}{m}\right\rceilVariables
- N: total number of objects
- m: number of containers
- n_i: occupancy of container i
How to use this formula
Guarantees that at least one of m containers holds at least the ceiling of N divided by m objects.
Important notes
- The principle guarantees existence but does not identify the container.
- Containers and assignments must be clearly defined.
Quick example
Placing 13 objects into 4 boxes guarantees one box contains at least 4 objects.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The principle guarantees existence but does not identify the container.
- For the Pigeonhole Principle Bound, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Define whether selections are ordered, repeated, labeled, or constrained before applying the counting formula.
Worked example
Placing 13 objects into 4 boxes guarantees one box contains at least 4 objects.
Common mistakes
- When copying Pigeonhole Principle Bound, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Verify the result of Pigeonhole Principle Bound with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Pigeonhole Principle Bound in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\max_i n_i\ge\left\lceil\frac{N}{m}\right\rceil. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Pigeonhole Principle Bound used for?
Guarantees that at least one of m containers holds at least the ceiling of N divided by m objects.
Can I copy this formula as LaTeX?
Yes. Copy \max_i n_i\ge\left\lceil\frac{N}{m}\right\rceil or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.