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Permutation and Combination
by Anupam Mazumdar - Monday, 27 October 2014, 08:06 PM
 
Finding numbers of shortest paths in a network of roads with restrictions. Please check the attachment. I really need to know how to solve it
Re: Permutation and Combination
by TG Team - Tuesday, 28 October 2014, 02:13 PM
 
Hi Anupam smile

Just observe that Peter can go along rightwards or upwards as he is to travel minimum possible distance. So just calculate the number of ways to reach any particular vertex as shown in the figure below and you will get the answer.

Also note that every vertex is connected with direct roads from exactly four other vertices {one upward, one downward, one leftward and one rightward}. Because Peter is to travel only in two particular directions so it becomes easier to calculate the number of ways to reach every vertex.

More questions on this concept and many other concepts too have been discussed at my article. Check this - Kamal Lohia's Math Corner. smile

Re: Permutation and Combination
by Anupam Mazumdar - Wednesday, 29 October 2014, 10:25 AM
  Yes..I can do that but the process will take a lot of time, specially for bigger dimensions. Is there any mathematical way?

The answer key was (10c4)-((5c2)x(5c2))-((6c2)x(4c2))+((5c2)x(4c2))

10c4 is the total number of ways going from x to y and 5c2 is the number of ways crossing the park. What about the other terms?

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