There are 12 towns grouped into four zones with three towns per zone. It is…

Concept: To count connections between grouped items where the connection rule differs by group membership, split every possible pair into two categories - pairs within the same group and pairs across different groups - using the combination count C(n, 2) = n(n - 1) / 2 for the number of pairs among n items. Apply the given rate of lines to each category separately, then add the two totals.

Application: There are 12 towns split into 4 zones of 3 towns each.

1. Within one zone, the number of town-pairs is C(3, 2) = 3.

2. Each same-zone pair needs 3 direct lines, so one zone needs 3 x 3 = 9 lines; across all 4 zones, the same-zone lines total 4 x 9 = 36.

3. The total number of pairs among all 12 towns is C(12, 2) = 66.

4. Subtracting the same-zone pairs (3 pairs x 4 zones = 12) from 66 leaves 66 - 12 = 54 pairs that lie in different zones.

5. Each different-zone pair needs exactly 1 direct line, so the different-zone lines total 54 x 1 = 54.

6. Adding both categories: 36 (same-zone) + 54 (different-zone) = 90 direct telephone lines.

Cross-check: Count independently from each town's point of view. Every town has 2 zone-mates, connected via 3 lines each, giving 2 x 3 = 6 line-ends from its own zone; it also has 9 towns outside its zone, connected via 1 line each, giving 9 more line-ends - a total of 6 + 9 = 15 line-ends per town.

Summing over all 12 towns gives 12 x 15 = 180 line-ends; since every line has two ends (one at each connected town), the actual number of distinct lines is 180 / 2 = 90, matching the total above.