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Direction for the question given below

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 Neelam resides 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

1. 60
2. 75
3. 45
4. 90
5. 72
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90 ## 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

by (266 points) 2 8
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The point why she should reach E is not clear..
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So that she can get diagonal path
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Yes, thats the most beautiful part of the answer - should be clear :)
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To avoid 4 roads(Two to right and Two to left), so as to make it minimum, we are bound to go through diagonal and so every path will contain EF in their route. The answer provided in the previous post is right but the explanation is incorrect.

The shortest path will be A → E → F→ B. Because the entire map is symmetric but the diagonal creates a new shorter path from E→ F. Diagonal is shorter than the sum of its perpendicular and base (Pythagoras Theorem’s Corollary).

Now to go from A to E, we have to travel right two times and travel downwards two times. So, we could for example, take R, D, R and D or we could take R, R, D, D. So how many such arrangements are possible ?? Lets list them down :-

R R D D, D D R R, R D R D, D R D R, R D D R, D R R D – Six arrangements possible

So the questions is how many permutations/arrangements of these four roads (R R D D) is possible given two of them are repeating twice.

It is : $\large \frac{^{4} P_{4}}{2!*2!} = \frac{4!}{2! * 2!} = 6$

Similarly, for F to B, we have to take four rights and two downs (R R R R D D). So the number of arrangements of these six roads, one which repeats twice and the other repeats four times is :

$\large \frac{^{6} P_{6}}{4!*2!} = \frac{6!}{4! * 2!} = 15$

Therefore, the total number of ways are :  $\large 6 *15 = 90$.

by (142 points) 1 3