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Method wanted

Consider an \(m\times n\) rectangular grid . Find the total number of paths one can reach from lower left corner to upper right corner .

Plz post ur method and other variations possible in such questions,

For example the number of shortest path possible from one corner to opposite corner is \(\frac{(m+n)!}{m!\times n!}\)

I dont know how to solve if its asked number of paths possible to reach from one corner to the corner above it.

Note by Tanishq Varshney
2 years, 2 months ago

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I think this problem would be more interesting if it asked for the number of ways to reach the opposite square, not being able to retrace your path, i.e. go on squares you have already been on. Andrei Golovanov · 2 years, 2 months ago

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@Andrei Golovanov right. I guess that's what meant Gian Sanjaya · 2 years, 2 months ago

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it's infinite if you said all path. Obviously, because you told all possible paths, and you can return to a point you started, that makes number of possible paths infinite. What's wrong? Gian Sanjaya · 2 years, 2 months ago

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Can you add your explanation to Rectangular Grid Paths wiki? Thanks! Calvin Lin Staff · 2 years, 2 months ago

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Assume person starts from lower left corner. To take the shortest path, one can travel only up or right in each step. And there are of course, \(m+n\) steps to take. In the end, the person is at the top right corner, this means that he/she has traveled \(m\) units up and \(n\) units right. The order in which these steps were arranged is the thing that matters here and is the thing we have to count. Basically you need the coefficient of \(x^ny^m\) in \((x+y)^{m+n}\). Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan well this set helps a lot Tanishq Varshney · 2 years, 2 months ago

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@Tanishq Varshney Yes, I saw that set.. My friend gave me similar qs.. so I din't go for solving them again. Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan if total number of possible paths are asked then?? Tanishq Varshney · 2 years, 2 months ago

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@Tanishq Varshney Then the answer is infinite, as one can keep going in loops around the grid. Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan ok, if one is allowed to move p steps noth and q steps east, then Tanishq Varshney · 2 years, 2 months ago

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@Tanishq Varshney I do not understand your question. How is it different from the one initially discussed in this note? Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan I mean to say if one has the condition to move 3 steps right and 2 steps up Tanishq Varshney · 2 years, 2 months ago

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@Tanishq Varshney It is still no different. If one is forced to move only three steps up, then one can take steps upward which are 3 blocks in size. This is same as saying that the person moves upwards one block when the total number of vertical blocks are divided by three.

eg: if there are thirty vertical blocks and person can take three steps only, this is same as 10 vertical blocks when the person is taking one step each. Raghav Vaidyanathan · 2 years, 2 months ago

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@Tanishq Varshney Can u post solution for the problems ants on a cube Tanishq Varshney · 2 years, 2 months ago

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