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In a city where all streets run east-to-west, all avenues run north-to

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Thoughtosphere
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In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   14 Oct 2014, 23:50
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In a city where all streets run east-to-west, all avenues run north-to-south, and all intersections are right angles as shown below, Jenn needs to walk from the corner of 1st Street and 1st Avenue to the corner of 6th Street and 3rd Avenue. If her friend Amanda is sitting on a bench on 4th Street halfway between 1st and 2nd Avenues, and Jenn chooses her path randomly from any route that will allow her to walk exactly seven blocks to her destination, what is the probability that Jenn will walk down 4th St. past Amanda?

A. 1/42
B. 1/21
C. 1/7
D. 1/3
E. 1/2

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In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   15 Oct 2014, 11:46
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Thoughtosphere wrote:

In a city where all streets run east-to-west, all avenues run north-to-south, and all intersections are right angles as shown below, Jenn needs to walk from the corner of 1st Street and 1st Avenue to the corner of 6th Street and 3rd Avenue. If her friend Amanda is sitting on a bench on 4th Street halfway between 1st and 2nd Avenues, and Jenn chooses her path randomly from any route that will allow her to walk exactly seven blocks to her destination, what is the probability that Jenn will walk down 4th St. past Amanda?

A. 1/42
B. 1/21
C. 1/7
D. 1/3
E. 1/2




Jenn needs to walk from red dot to green dot in seven moves. So, she should walk 5 blocks UP and 2 blocks LEFT: UUUUULL. She can do this in 7!/(5!2!) = 21 ways, which is the number of permutations of 5 U's and 2 L's.

How many roots go through the black dot?
To walk from red to blue Jenn needs to walk 3 blocks UP: UUU. So, 1 way.
To walk from yellow to green Jenn needs to walk 2 blocks UP and 1 blocks LEFT: UUL. She can do this in 3!/2! = 3 ways.

So, only 1*3 = 3 ways pass through black dot.

Probability = 3/21 = 1/7.

Answer: C.

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Re: In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   15 Oct 2014, 05:52
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All routs between (1,1) to (3,6) = 7!/(2!*5!) = 21
Routs which pass 4th st. between 1st and 2nd avenue = routs between (1,1) to (1,4) * routs between (2,4) to (3,6)
= 1 * (3!/(2!*1!)) = 3
the probability that Jenn will walk down 4th St. past Amanda= 3 / 21 = 1 / 7

C is correct.
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Re: In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   30 Dec 2017, 01:00
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UmangMathur wrote:

In a city where all streets run east-to-west, all avenues run north-to-south, and all intersections are right angles as shown below, Jenn needs to walk from the corner of 1st Street and 1st Avenue to the corner of 6th Street and 3rd Avenue. If her friend Amanda is sitting on a bench on 4th Street halfway between 1st and 2nd Avenues, and Jenn chooses her path randomly from any route that will allow her to walk exactly seven blocks to her destination, what is the probability that Jenn will walk down 4th St. past Amanda?

A. 1/42
B. 1/21
C. 1/7
D. 1/3
E. 1/2

Show Spoiler
Attachment:
Street_Map.png


VERITAS PREP OFFICIAL SOLUTION:

C. Given the diagram, you should see that Jenn will make it to her destination by traveling 5 "up" blocks (from 1st to 6th St.) and 2 "left" blocks (from 1st to 3rd Ave.). How can you find all the routes? Using permutations - how many ways would you arrange 5 up and 2 left?
7!/(5!2!), which equals 21. Then to pass Amanda, she has to go up 1st Ave. to 4th St. and make her first left there, leaving 2 "up" streets and 1 "left" street left to go after she passes Amanda. That's 3 ways of the 21 (LUU, ULU, UUL) that will take her past Amanda, for an answer of 1/7.
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Re: In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   15 Oct 2014, 08:09
hi srezar,

i get ur approach..but can u explain graphically how u found the favourble outcomes(Routs which pass 4th st. between 1st and 2nd avenue = routs between (1,1) to (1,4) * routs between (2,4) to (3,6)
= 1 * (3!/(2!*1!)) = 3)??

-h
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Re: In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   15 Oct 2014, 12:13
Bunuel wrote:
Thoughtosphere wrote:

In a city where all streets run east-to-west, all avenues run north-to-south, and all intersections are right angles as shown below, Jenn needs to walk from the corner of 1st Street and 1st Avenue to the corner of 6th Street and 3rd Avenue. If her friend Amanda is sitting on a bench on 4th Street halfway between 1st and 2nd Avenues, and Jenn chooses her path randomly from any route that will allow her to walk exactly seven blocks to her destination, what is the probability that Jenn will walk down 4th St. past Amanda?

A. 1/42
B. 1/21
C. 1/7
D. 1/3
E. 1/2


Attachment:
Untitled.png

Jenn needs to walk from red dot to green dot in seven moves. So, she should walk 5 blocks UP and 2 blocks LEFT: UUUUULL. She can do this in 7!/(5!2!) = 21 ways, which is the number of permutations of 5 U's and 2 L's.

How many roots go through the black dot?
To walk from red to blue Jenn needs to walk 3 blocks UP: UUU. So, 1 way.
To walk from yellow to green Jenn needs to walk 2 blocks UP and 1 blocks LEFT: UUL. She can do this in 3!/2! = 3 ways.

So, only 1*3 = 3 ways pass through black dot.

Probability = 3/21 = 1/7.

Answer: C.

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Thanks a lot Bunuel, I got his one.

While searching for explanations, I came across another explanation.

The destination can be reached in minimum steps by going 5 up and 2 L.
Thus from the 7 choices, we need to choose how can we select the 5 Ups or the two downs

So, it would be \(7C5 or 7C2 = 7! / (5! * 2!)\)

Thanks
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Re: In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   15 Oct 2014, 12:16
Bunuel, Thanks a lot for sharing the similar types of questions, they come as life savers - help me understand the topic well... :-)
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In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   07 Jan 2018, 16:24
Bunuel wrote:
UmangMathur wrote:

In a city where all streets run east-to-west, all avenues run north-to-south, and all intersections are right angles as shown below, Jenn needs to walk from the corner of 1st Street and 1st Avenue to the corner of 6th Street and 3rd Avenue. If her friend Amanda is sitting on a bench on 4th Street halfway between 1st and 2nd Avenues, and Jenn chooses her path randomly from any route that will allow her to walk exactly seven blocks to her destination, what is the probability that Jenn will walk down 4th St. past Amanda?

A. 1/42
B. 1/21
C. 1/7
D. 1/3
E. 1/2

Show Spoiler
Attachment:
Street_Map.png


VERITAS PREP OFFICIAL SOLUTION:


C. Given the diagram, you should see that Jenn will make it to her destination by traveling 5 "up" blocks (from 1st to 6th St.) and 2 "left" blocks (from 1st to 3rd Ave.). How can you find all the routes? Using permutations - how many ways would you arrange 5 up and 2 left?
7!/(5!2!), which equals 21. Then to pass Amanda, she has to go up 1st Ave. to 4th St. and make her first left there, leaving 2 "up" streets and 1 "left" street left to go after she passes Amanda. That's 3 ways of the 21 (LUU, ULU, UUL) that will take her past Amanda, for an answer of 1/7.


Hi Bunuel,

I solved this problem the same way you did but I got hung up at the end. The portion of the question that says "what is the probability that Jenn will walk down 4th St. past Amanda?' confused me for a second because I mapped out the path as being NNNW which came out to 4!/3!. That would have resulted in an answer of 4/21. Do we assume that she takes the left, knowing that she is really moving half the distance to the left? If the question read "If her friend Amanda is sitting on a bench on 4th Street and 2nd Avenue" would 4/21 have been correct?
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Re: In a city where all streets run east-to-west, all avenues run north-to [#permalink] New post   07 Jan 2024, 09:00
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Re: In a city where all streets run east-to-west, all avenues run north-to [#permalink]
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