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by EricKryk » Thu Feb 20, 2014 8:32 am
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The diagram above shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?

A) 6
B) 7
C) 12
D) 14
E) 17
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by Patrick_GMATFix » Thu Feb 20, 2014 9:00 am
Moving from X to Y is equivalent to going through a series of decision points. Every time we have two options, the number of paths will double. Every time we have three options, the number of paths will triple...

The full solution below is taken from the GMATFix App.

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by Brent@GMATPrepNow » Thu Feb 20, 2014 9:15 am
EricKryk wrote:Image
The diagram above shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?

A) 6
B) 7
C) 12
D) 14
E) 17
Image

First recognize that, in order to get from point X to point Y, we MUST travel through points A,B,C,D,E and F.

So, we can take the task of getting from point X to Y and break it into stages.

Stage 1: Move from point X to point A
There's only 1 possible route, so we can complete stage 1 in 1 way.

Stage 2: Move from point A to point B
There are 2 possible routes, so we can complete stage 2 in 2 ways.

Stage 3: Move from point B to point C
There's only 1 possible route, so we can complete stage 3 in 1 way.

Stage 4: Move from point C to point D
There are 2 possible routes, so we can complete stage 4 in 2 ways.

Stage 5: Move from point D to point E
There's only 1 possible route, so we can complete stage 5 in 1 way.

Stage 6: Move from point E to point F
There are 3 possible routes, so we can complete stage 6 in 3 ways.

Stage 7: Move from point F to point Y
There's only 1 possible route, so we can complete stage 7 in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus move from point X to point Y) in (1)(2)(1)(2)(1)(3)(1) ways ([spoiler]= 12 ways[/spoiler])

Answer: C

Cheers,
Brent

Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775
Brent Hanneson - Creator of GMATPrepNow.com
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by GMATGuruNY » Thu Feb 20, 2014 12:33 pm
EricKryk wrote:Image
The diagram above shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?

A) 6
B) 7
C) 12
D) 14
E) 17
Number of ways to travel across the leftmost diamond = 2. (Either of the 2 paths.)
Number of ways to travel across the middle diamond = 2. (Either of the 2 paths.)
Number of ways to travel across the rightmost diamond = 3. (Any of the 3 paths.)
To combine these options, we multiply:
2*2*3 = 12.

The correct answer is C.
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by Jeff@TargetTestPrep » Mon Jun 29, 2015 6:59 am
EricKryk wrote:Image
The diagram above shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?

A) 6
B) 7
C) 12
D) 14
E) 17
Solution:

Image

A good way to solve this problem is to use the idea of the fundamental counting principle. In a more standard form you could be asked a question, such as if Tom as 3 belts, 4 ties, and 6 shirts, how many outfits could he make with those items? We can consider each item a decision point, i.e., belts, ties, and shirts. To solve this, we just need to multiply the number of decisions Tom can make together, so:

3 x 4 x 6 = 72 ways.

Tom has 72 options when dressing with those items.

The same logic can be applied to this problem. We first determine the number of ways the mouse can go from one point to the next. Notice we added in "decision points" to our diagram.

X to P1 = 1

P1 to P2 = 2

P2 to P3= 1

P3 to P4= 2

P4 to P5= 1

P5 to P6 = 3

P6 to Y = 1

Therefore, to determine the total number of ways from X to Y, we multiply all these numbers together:

1 x 2 x 1 x 2 x 1 x 3 x 1 = 12 ways.

There are 12 different paths.

Answer:C

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by stephanieh » Mon Jun 29, 2015 9:58 pm
GMATGuruNY wrote:
EricKryk wrote:Image
The diagram above shows the various paths along which a mouse can travel from point X, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from X to Y can the mouse take if it goes directly from X to Y without retracing any point along a path?

A) 6
B) 7
C) 12
D) 14
E) 17
Number of ways to travel across the leftmost diamond = 2. (Either of the 2 paths.)
Number of ways to travel across the middle diamond = 2. (Either of the 2 paths.)
Number of ways to travel across the rightmost diamond = 3. (Any of the 3 paths.)
To combine these options, we multiply:
2*2*3 = 12.

The correct answer is C.
Awesome break down Mitch! Thank you. I don't know how you're able to make it so simple! Do you have a thought process system as you're taking in the question?
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by nikhilgmat31 » Thu Jul 02, 2015 10:50 pm
please attached the image.
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