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Problem 1111 · The Empty ArrangementPrint one blank worksheet now, download its PDF, or prepare a family or classroom packet.

Problem 1111

The Empty Arrangement

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The Empty Arrangement

Complete to collectIndicoliteLevel 3 · 5 Glints

Meditations on Oneness

Local folio mark · 1111

Page 1 · Inspect the case with nothing in it

Three Gallery plates

The Gallery is preparing one plate for two seals, one for a single seal, and one for none. A complete arrangement of distinct seals places all seals into numbered cases, one seal per case, using every seal.

Let be the number of complete arrangements.

Three Gallery panels show two seals above two numbered cases, one seal above one numbered case, and an intentionally empty gold frame labeled zero seals and zero cases. Blank ledger lines invite arrangement records.
Use the first two panels to list arrangements. Decide whether the untouched third panel is complete.

The untouched display

Write arrangement lists on the diagram. Use scratch paper for the proof.

  1. List every complete arrangement of two seals in two numbered cases. Count the complete arrangements of one seal in one case. Now consider zero seals and zero cases. Does the untouched display count as a complete arrangement? Prove that there is exactly one complete arrangement—not zero arrangements and not more than one.

____ ____ ____

Page 2 · Follow the empty echo

Two independent checks

Choose the first case

For , choose one of the seals for case 1, then arrange the remaining seals.

Record arrows

A permutation of a collection sends every object to one destination in , with no repeated or missing destinations.

Check the count twice

Use separate paper for both proofs.

  1. Use the first-case choice to justify for . Substitute to determine . Explain why this agrees with Part 1. Compare your result with this rule: a product containing no factors has value .

  2. The identity permutation sends every object to itself. Prove that every collection has exactly one identity permutation. Describe its instruction list when is empty.

Nothing is placed. Nothing is omitted. One arrangement is complete.

At a glance

About this problem

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Theme

Skills

Grade

Math area

Topics

Is this a good fit?

Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 8 · Grade 9 · Grade 10

Uses HS DISC math.

Thinking

Stretch challenge

A nontrivial plan using familiar mathematical ideas.

Time & response

About 12–28 minutes

Moderate amount of written work · Proof response.

How to support: The useful question is not whether anything is visible, but whether every arrangement rule is satisfied. Ask how a second empty arrangement could differ from the first.

How did it feel?

Pick the option that best matches this challenge. Recommendations use topic, math area, grade, story, and thinking challenge.

Keep exploring

The Manyfold challenge 18 of 22

Technical reference

Rating, skills & curriculum

Designed for Grade 8 · Grade 9 · Grade 10 · Uses HS DISC math · About 12–28 minutes

Grade 8 · Grade 9 · Grade 10 · Discrete mathematics and logic · Combinatorics

MJ HS:DISC.2 · C3 · W2

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ HS:DISC.2

High school

HS DISC · point 2

Likely range
MJ HS:DISC.1–HS:DISC.3
Confidence
86%

Challenge

C3 · Stretch

A nontrivial plan using familiar mathematical ideas.

  • Method selection2
  • Reasoning depth2
  • Novelty3
  • Constraint management2
  • Proof or justification3

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
12–28 minutes · typically 19
Response
Proof
Known-method steps
13
  • Mechanical execution1
  • Written output3
  • Representation production2
  • Bookkeeping1

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Count permutations and combinations with constraints

    The learner can count permutations and combinations with constraints. This capability is distinct from mere exposure to or isolated use of factorials; counting principle.

    • Developing mastery
    • 100% focus
    • MJ HS:DISC.2 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Common Core State Standards for Mathematics2010 · 1 location
Standards1 location · Representative: (+) Use permutations and combinations to compute probabilities of compound events and solve problems.