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

Question

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?

(A) 72
(B) 96
(C) 90
(D) 140
(E) 150

(A) 72
(B) 96
(C) 90
(D) 140
(E) 150

KarishmaB · Accepted Answer

Consider a town A. It has 3 direct lines each with the other two towns in its zone (so 6 lines) and 9 direct lines with the 9 towns out of its zone. 
So A has 15 lines. Since there are 12 towns, we have 15*12 lines but here, each line is counted twice (since it connects 2 towns).
Hence, actual number of lines is 15*12/2 = 90 lines.

Answer (C)

JohnGalt1 · Answer

Lets consider =>

=> Firstly, we need to calculate for same zone:
The 3 towns in a zone can be connected to each other with 3*3=9 lines in each zone.
Total of 4 zones will require 9*4=36 lines.

=> Secondly, we calculate for different zones
Now total ways of choosing 2 towns amongst 12 towns = 12C2=66. But we shouldn't forget of these 3C2*4 or 12 which are in the same zone, so we substract 66-12 or and we get that 54 are in different zones.
As each such way has only one line, total lines = 54*1=54 lines..

IF we sum up two stages we get 36+54=90 lines

Thus C should be correct

ScottTargetTestPrep · Answer

Solution:
 
All 12 towns, regardless if they are in the same zone or not, need to be connected with only one direct line with each other, so they need 12C2 = (12 x 11) / 2 = 66 direct lines.

However, since the towns within the same zone need to connect to one another with three direct lines (i.e., 2 additional lines), each zone requires 3C2 x 2 = 6 more direct lines and thus, 4 zones require 6 x 4 = 24 more direct lines.

Therefore, the total number of direct lines needed is 66 + 24 = 90.

Answer: C

Sanskar115 · Answer

Given:
- 12 towns are grouped into four zones with three towns per zone.
- 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.
to prove: 
- How many direct telephone lines are required?
Solution: 
telephone lines to the same zone = 6, as two towns are connected with three direct lines, 3×2
total telephone lines = 11 +10+ 9+8+7+6+5+4+3+2+1 +6×4, ...((6×4) for the 4 zones, and the sum of 1 to 11 as inter-zone are connected with one direct line to each other
=> \frac{11}{2}×(1+11)+24 
=>66 +24 = 90

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

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

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