CAT 2003 Slot 2 — QA Question 34
There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required?
Answer & solution
Correct answer: 90
- A
72
90
- C
96
- D
144
Case 1: Lines within Zones
Within every zone there will be 3 + 3 + 3 = 9 lines
∴ Total number of phone lines within each zone for all 4 zones together = 9 × 4 = 36
Case 2: Lines connecting different Zones
Let Zone 1 have towns A, B and C.
A will be connected to 9 towns of the other 3 zones, each through a single direct line.
Similarly, B and C will also be connected to 9 different towns.
∴ The number of direct lines from Zone 1 to other towns (in Zones 2, 3 and 4) = 3 × 9 = 27
For Zone 2, we must only count the number of lines to Zones 3 and 4 (lines between Zone 1 and Zone 2 have already been considered.)
∴ The number of lines from the 3 towns of Zone 2 to other towns (in Zones 3 and 4) = 3 × 6 = 18
Now, for Zone 3, we should only count the number of lines to Zone 4 (lines to Zones 1 and 2 have already been considered.)
∴ The number of direct lines from the 3 towns of Zone 3 to the 3 towns in Zone 4 = 3 × 3 = 9
∴ Total number of direct telephone lines = 36 + 27 + 18 + 9 = 90
Hence, option (b).