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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.  [#permalink]

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New post 12 Nov 2019, 02:55
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Question Stats:

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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) 90
(C) 96
(D) 144
(E) 120


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Re: There are 12 towns grouped into four zones with three towns per zone.  [#permalink]

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New post 12 Nov 2019, 10:00
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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


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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
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Re: There are 12 towns grouped into four zones with three towns per zone.  [#permalink]

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New post 13 Nov 2019, 05:01

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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Re: There are 12 towns grouped into four zones with three towns per zone.  [#permalink]

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New post 20 Nov 2019, 18:53
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 Questions



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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Re: There are 12 towns grouped into four zones with three towns per zone.   [#permalink] 20 Nov 2019, 18:53
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