# Practice files A–F · entirely fictional data

These small finite sets make every count checkable by regions or a tree. “Both” means the intersection; with three sets, a pairwise intersection **includes** anyone in all three. The same person or option is never counted twice in a union. Contexts are examples, not collected learner data.

## A · Library makerspace interests

Out of **30** fictional visitors, **17** chose sketching (S), **13** chose coding (C), and **5** chose both. Thus `|S∪C| = 17+13−5 = 25`; **5** chose neither. Region check: S only **12**, both **5**, C only **8**, neither **5**; `12+5+8+5=30`. Do not say all 30 chose an activity.

## B · Find the unknown overlap

In a fictional **40**-person workshop, **22** chose drawing (D), **18** chose music (M), and **32** chose at least one. Hence `|D∩M| = 22+18−32 = 8`. Neither = `40−32 = 8`. Regions: D only **14**, both **8**, M only **10**, neither **8**.

## C · Three clubs, with explicit pair meanings

In a fictional **28**-person week: art A **12**, robotics R **11**, choir C **9**; `|A∩R|=4`, `|A∩C|=3`, `|R∩C|=5`, and `|A∩R∩C|=2`. Every pair count includes the two people in all three. The union is `12+11+9−4−3−5+2 = 22`; **6** are in none. Exact regions: only A **7**, only R **4**, only C **3**, A&R only **2**, A&C only **1**, R&C only **3**, all three **2**. Sum **22**. No region is negative, and every set total reconstructs.

## D · One session, not two

A fictional afternoon programme lets a participant book **one** of three distinct studio sessions **or one** of four distinct outdoor sessions. The two booking categories are mutually exclusive in this choice. Addition gives **3+4=7** possible bookings. It is not 3×4 because a person books exactly one session. If a session appears in both lists, the overlap must first be removed.

## E · Two independent stages

A fictional display label has **3** shape choices and **4** caption choices. Every shape works with every caption. One finished label contains one shape **and** one caption, so multiplication gives **3×4=12** labels. This is not a permutation of the seven items; the stages have different roles.

## F · Branch, then multiply within each branch

A fictional community event offers **one** of two exclusive delivery routes. A visual card has **2** theme choices and **3** layouts: **6** cards. An audio card has **2** narration choices and **2** pacing choices: **4** cards. One chooses visual **or** audio, never both in this model: **2×3 + 2×2 = 10** routes. The case makes no claim that a particular mode suits a particular kind of learner; these are optional presentation choices.