Two cases..

(I) within SAME zone..
The 3 towns in a zone can be connected to each other in 3C2 or 3 ways. Each combination has 3 lines, so 3*3=9 lines in each zone.
4 such zones will require 9*4=36 lines.

(II) Between DIFFERENT zones..
Now total ways of choosing 2 towns amongst 12 towns = 12C2=66. Out of these 3C2*4 or 12 are within SAME zone, so 66-12 or 54 ways are in DIFFERENT zones.
As each such way has only one line, total lines = 54*1=54 lines..

Total = case I + case II = 36+54=90 lines

B

Solution

Given

• 12 towns grouped into four zones with three towns per zone.
• Every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise

To find

• The number of direct telephone lines required.

Approach and Working out

Total number of lines within same zone

• Every two towns are connected by 3 lines. So, total lines = Total ways of selecting two towns * 3 = 3c2 * 3 = 9
• Total direct lines in 4 zones = 9*4= 36

Total number of lines that are not in the same zone

• Total ways of selecting two towns out of 12 towns * 1 = 12c2 * 1 = 66
• Out of these 66 lines, 12 lines will be in the same zone that we double-counted.

o Hence, 66-12= 54 lines are there between towns that are not in the same zone.

Total lines = 36 + 54 = 90 lines

Thus, option B is the correct answer.

Correct Answer: Option B
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Within a single zone, there are 3C2 = 3!/(2!*1!) = 3 ways to choose two cities. Thus, within one zone, 3 * 3 = 9 telephone lines are needed, and since there are four zones, 9 * 4 = 36 lines are needed to connect cities within the same zone.

Notice that there are a total of 12C2 = 12!/(2!*10!) = (12 x 11)/2 = 66 ways to choose two cities, including cities that are within the same zone. We calculated above that there are 3 ways to choose two cities within a single zone and since there are four zones, there are a total of 3 * 4 = 12 ways to choose two cities within the same zone. Since the total number of choices is 66, there are 66 - 12 = 54 ways to choose two cities within different zones. So, 54 lines are needed to connect cities within different zones.

In total, 36 + 54 = 90 lines are needed.

Answer: B
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