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Two highways start from a point P and meet again...

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Two highways start from a point P and meet again at the point Q. Between P and Q there are 5 subways, joining the highways at 5 different places. How many different routes are possible for a journey from P to Q?

(A) 12
(B) 16
(C) 24
(D) 32
(E) 64


not sure, how to work with this kind of problem
[Reveal] Spoiler: OA

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Re: Two highways start from a point P and meet again... [#permalink]

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mynhauzen wrote:
Two highways start from a point P and meet again at the point Q. Between P and Q there are 5 subways, joining the highways at 5 different places. How many different routes are possible for a journey from P to Q?

(A) 12
(B) 16
(C) 24
(D) 32
(E) 64


not sure, how to work with this kind of problem


No. of ways of using no subways = \(2*5C_0 = 2*1 = 2\)

No. of ways of using one subway = \(2*5C_1 = 2*5 = 10\)

No. of ways of using two subways = \(2*5C_2 = 2*10 = 20\)

No. of ways of using three subways = \(2*5C_3 = 2*10 = 20\)

No. of ways of using four subways = \(2*5C_4 = 2*5 = 10\)

No. of ways of using five subways = \(2*5C_5 = 2*1 = 2\)

We are multiplying by 2 on each occasion because for each combination, we can start the trip either by route 1 or by route 2.

Total = 2 + 10 + 20 + 20 + 10 + 2 = 64
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Re: Two highways start from a point P and meet again... [#permalink]

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New post 14 Feb 2013, 02:23
If one starts its journey from point P, one has 2 options ( to go via either highway)... When he reaches 1st subway point he has again, 2 options, to continue on the highway or to go through subway., When he reaches next subway, he will have same two options, of course we are considering that he doesn't return from any point.
Since, there are 5 subways, he has the option of choosing from 2 choices 6 times (one when he just starts his journey). So, total ways he can reach is destination is 2^6 = 64.

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mynhauzen wrote:
Two highways start from a point P and meet again at the point Q. Between P and Q there are 5 subways, joining the highways at 5 different places. How many different routes are possible for a journey from P to Q?

(A) 12
(B) 16
(C) 24
(D) 32
(E) 64


not sure, how to work with this kind of problem


This is a poor quality question. In order the answer to be 64, must be mentioned that we should go directly from P to Q without retracing any point along a path, otherwise infinitely many routs will be possible.

In this case, along the path from P to Q we'll have 6 forks and 2 options for each (starting from P), thus there are 2^6=64 routs possible.

Answer: E.

Proper question testing the same concept is here: the-diagram-above-shows-the-various-paths-along-which-a-mous-144271.html

Hope it helps.
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Re: Two highways start from a point P and meet again... [#permalink]

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New post 13 May 2016, 09:08
Option E it is .. well explained above !
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Re: Two highways start from a point P and meet again...   [#permalink] 13 May 2016, 09:08
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