Ratish, imagine that the region AKML is blocked off. Now if you have to go from A towards B you would have two options to start off, either via K or via L. Note also that you cannot visit K _and_ L and still manage a “shortest-path” route. So we can safely say “either K or L”. From the Fundamental principle of Counting (see https://crackthecat.wordpress.com/2013/03/25/the-fundamental-principle-of-counting/), we know that this means we can add up “number of ways via K” and “number of ways via L”

Number of ways to get to K from A is just 1. So we need to only count ways from K to B, which require 11 steps, 6 horizontal and 5 vertical, which can be done (as in the previous post) in 11C5 ways. Similarly one can go from A to L in only one way, and from there onward to B in 11C3 ways (think about it!) hence we can say 11C5 + 11C3

For (g), the question should be “How many paths are there from A to B?” and not shortest path. This is because in order to reach B, starting from M will only give shortest way. Am I going wrong somewhere?

No, there could be “shortest paths” which do not pass through M at all (along the boundary of the rectangle, for example). And if I remove the shortest criterion there could be many many longer paths….

Can you please explain the answer for G. Can you explain the roads considered? Please be a little more elaborate

Ratish, imagine that the region AKML is blocked off. Now if you have to go from A towards B you would have two options to start off, either via K or via L. Note also that you cannot visit K _and_ L and still manage a “shortest-path” route. So we can safely say “either K or L”. From the Fundamental principle of Counting (see https://crackthecat.wordpress.com/2013/03/25/the-fundamental-principle-of-counting/), we know that this means we can add up “number of ways via K” and “number of ways via L”

Number of ways to get to K from A is just 1. So we need to only count ways from K to B, which require 11 steps, 6 horizontal and 5 vertical, which can be done (as in the previous post) in 11C5 ways. Similarly one can go from A to L in only one way, and from there onward to B in 11C3 ways (think about it!) hence we can say 11C5 + 11C3

regards

J

6 horizontals and 4 verticals from K to B. So should not it be 10C6 sir ?

There are 5 verticals!

regards

J

Should not it be 10C4 since there are 6 horizontals and 4 verticals. 10steps from K to B??,

how can we solve (g) with diagonal method u gave in post 4 ??

Sir,

In problem G:

Can there be one more way, A to M in 2 ways and M to B in 9C3 ways?

Sorry i think the path which i am talking about already gets covered in the two as mentioned by you.

For (g), the question should be “How many paths are there from A to B?” and not shortest path. This is because in order to reach B, starting from M will only give shortest way. Am I going wrong somewhere?

No, there could be “shortest paths” which do not pass through M at all (along the boundary of the rectangle, for example). And if I remove the shortest criterion there could be many many longer paths….

regards

J

What if i reach M first and then take a short path then the number of ways 2C1*9C3

There are “shortest paths” which do not pass through M….see previous response!

regards

J