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12 Mar 2021, 03:16
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C
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Difficulty:
85%
(hard)
Question Stats:
53% (03:01) correct
47%
(02:40)
wrong
based on 94
sessions
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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?
(A) 72
(B) 96
(C) 90
(D) 140
(E) 150
(A) 72
(B) 96
(C) 90
(D) 140
(E) 150
28 Apr 2022, 00:39
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Bunuel
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
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)
General Discussion
JohnGalt1
Joined: 10 Feb 2021
Last visit: 13 Apr 2022
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12 Mar 2021, 03:49
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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
=> 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
