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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?

1. $72$
2. $90$
3. $96$
4. $144$

There are 4 zones of 3 towns each.

In a zone there are 3 towns.

Each town connects with the other town in 3 lines.

Hence total number of lines is 3* 3 = 9.

4 zones have 9*4 = 36 direct links within it.

Now,

Consider a town in Zone 1. There are 9 towns outside this zone.

Hence 9 lines are required to connect between this town and the other 9 towns.

Similarly, there are 12 towns, each with 9 towns outside its own zone.

So, totally, there will be 9*$\frac{12}{2}$ = 9* 6 = 54 lines inter zone

So, the total number of lines will be 36 (within zone) + 54 ( inter zone) = 90.

Hence Option (B)90 is the Answer.

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