Book Isle counting problems: distinguish selecting two people from assigning two roles

A problem mentioning four people and two choices can yield six or twelve depending on what counts as an outcome. Read an original group exercise in Book Isle, list unordered pairs and then assign explainer and recorder roles. Enumeration is manual; the app does not identify combinations or automatically solve a counting problem from headcounts.

1. Fix people, roles and repetition rules

Prepare an original exercise with four distinct people A, B, C and D. Each round chooses two different people. Record file, edition, chapter/page and a phrase anchor, keeping group selection separate from role assignment. Question one asks only who joins, without order. Question two assigns distinct explainer and recorder roles, one per person. Different assumptions about repeated choices or identities require another model. Use available notes or bookmarks, or transcribe the necessary statement manually. No automatic parsing or completion of unstated rules is assumed.

2. Give unordered groups a unique notation

Write every unordered group in alphabetical order. AB and BA represent the same group in question one. Enumerate AB, AC, AD, then BC, BD, then CD: six groups. Alphabetical notation prevents duplicates; it does not make A more important. Check distinct members and coverage, excluding AA and BB. This recording order does not secretly assign explainer and recorder roles. State what an outcome means before deciding whether two records should merge.

3. Explain the division by two using ordered choices

An alternative first chooses one of four people, then one of the remaining three: twelve ordered choices. For the role-free question, every group occurs exactly twice, such as AB and BA, so divide by two to obtain six. The permutation/combination distinction in OpenStax counting principles depends on outcome identity. Division comes from this duplication, not merely from the phrase “two people.” If roles distinguish exchanges, do not divide those arrangements away. Pair the twelve records into six groups to explain the operation and exclude repeated-person cases.

4. Assign two different roles within each group

For question two use the explicit format explainer, recorder. Group AB yields A explaining/B recording and B explaining/A recording: the people match but their role mappings differ. Give both arrangements for every group, obtaining twelve. Regroup by explainer and check three possible recorders for each person. Do not erase exchanged arrangements using the first question’s alphabetical notation, or add one-person dual-role outcomes. Keep group identity alongside both roles so a reader can trace the factor of two rather than guessing why the total became twelve.

5. Compare individual outcomes to expose mixed models

Take AC and write its unordered group and both role assignments. Cover the totals and ask whether exchanging people gives the same outcome. Yes for question one; no for question two, by definition rather than visual name order. Two undifferentiated co-recorders do not become ordered because you wrote A first. Conversely, similar workloads do not merge explicitly distinct roles. Retain role names instead of flattening both into participants. For an ambiguous statement, give condition-dependent candidate counts rather than replacing the ambiguity with a familiar number.

6. Recount when the rules change

In one control exclude D. The remaining unordered pairs are AB, AC and BC; distinct-role assignments total six. Keep the four-person original so the repeated number six does not conceal a changed model. In another control allow one person to hold both roles: four choices for each role give sixteen, including AA and three other repeated-person cases. This relaxes a rule rather than repairing an omission in the original. Every restriction or relaxation needs its own outcome definition and enumeration; equal totals from different questions do not make their processes identical.

7. Troubleshoot duplicate records and probability assumptions

Twelve unordered records suggest AB/BA duplicates. Six role assignments suggest exchanged duties were wrongly merged. Sixteen under a no-dual-role rule suggests AA, BB, CC and DD slipped in. Counts alone do not establish real drawing probabilities; equal-likelihood assumptions would be an additional requirement. This lesson does not judge real allocation fairness. Open copied or exported notes and inspect role columns, distinct-person rules and control versions, not just totals. Return through chapter and phrase when necessary rather than deleting a book to resolve a logic error.

8. Finish by enumerating again from the statement

Exit and return to the original problem. Without looking at a formula, enumerate six groups and write both role assignments for any two groups. Explain the D-exclusion and dual-role controls without overwriting original rules. Save edition/location, identities, selection rules, outcome meaning, enumeration and duplicate explanation. Completion means another reader can say what six and twelve count, not merely select a formula whenever two people appear. Rebuild the model for different people, roles or repetition rules; reading and notes do not replace that judgement.

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Sets and universal scope, Exclusive and independent events.

Checked on November 11, 2026 against the official Book Isle App Store listing and Apple public lookup: Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Use existing reading, available notes or external records and verify controls in your installed version. Original samples and manual calculations are not new automatic analysis features. No physical-device, private-file or universal export-environment testing is claimed.