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12 Nov 2019, 02:55
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
48% (02:56) correct
52%
(02:44)
wrong
based on 247
sessions
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Not Attempted Yet
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 Questions
(A) 72
(B) 90
(C) 96
(D) 144
(E) 120
Are You Up For the Challenge: 700 Level Questions
12 Nov 2019, 10:00
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Bunuel
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
General Discussion
13 Nov 2019, 05:01
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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
