Bunuel wrote:
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) 90
(C) 96
(D) 144
(E) 120
Are You Up For the Challenge: 700 Level QuestionsWithin 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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