CAT 2008 — QA Question 5
Directions for next 2 questions:
The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.
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Neelam rides her bicycle from her house at A to her office at B, taking the shortest path. Then the number of possible shortest paths that she can choose is
Answer & solution
- A
60
- B
75
- C
45
90
- E
72
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We can find the number of shortest possible paths from A to E either by trial and error or by using combinations.
Note that to travel from A to E, we have to take 2 roads to the right and 2 roads downwards (in the diagram) in order that we follow the shortest path. In other words, we have to use 2 + 2 = 4 roads, out of which 2 are towards right and 2 are downwards.
This is equivalent to selecting 2 things (roads towards right) out of 4 things (roads). (The remaining two roads will be downwards.)
The number of ways of doing this is 4C2 = 4!/(2!×2!) = 6
∴ From point A to E, there are 6 ways to reach with the minimum distance travelled.
Here E to F is the shortest distance because the third side of a triangle is always less than the sum of the other two sides.
From point F to B, there are 6C4 = 6!/(4!×2!) = 15 ways to reach with the minimum distance travelled.
∴ There are 15 × 6 = 90 shortest paths that Neelam can choose.
Hence, option (d).