There are 12 towns grouped into four zones, with three towns in each zone.…
2023
There are 12 towns grouped into four zones, with three towns in each zone. Telephone lines are to be laid so that every pair of towns is connected by three direct lines if they belong to the same zone, and by exactly one direct line if they belong to different zones. How many direct telephone lines are required in total?
- A.
96
- B.
114
- C.
72
- D.
90
Show answer & explanation
Correct answer: D
Concept
When n items are split into groups and every unordered pair of items is joined by a link, the total number of pairs is C(n,2). Splitting the total into pairs inside the same group and pairs across different groups lets you compute a mixed connection count: total links = (same-group pairs × rule for same group) + (cross-group pairs × rule for cross group), where cross-group pairs = C(n,2) minus same-group pairs.
Application
Count towns per zone and total towns: 4 zones × 3 towns = 12 towns.
Same-zone pairs in one zone: choosing 2 of 3 towns gives C(3,2) = 3 pairs; across 4 zones that is 4 × 3 = 12 same-zone pairs.
Same-zone lines: each such pair needs 3 direct lines, so 12 × 3 = 36 lines.
Total pairs among all 12 towns: C(12,2) = (12 × 11) / 2 = 66 pairs.
Cross-zone pairs: 66 total pairs − 12 same-zone pairs = 54 pairs.
Cross-zone lines: each such pair needs exactly 1 direct line, so 54 × 1 = 54 lines.
Total direct telephone lines: 36 + 54 = 90.
Cross-check
View it town-by-town instead: each town has 2 other towns in its own zone (3 lines apiece = 6) and 9 towns in other zones (1 line apiece = 9), giving 15 line-ends per town. Summing over all 12 towns counts every line from both its ends, so the true total is (12 × 15) / 2 = 90 — the same figure, confirming the result.