How to count the number of possible outcomes?

Question

Mathematics

There is a board M with N squares. It is possible to walk endlessly, but only down or to the right. You need to go from the top left to the bottom right
Task: count how many possible outcomes there are
Example of a board:
1 1 1 1
1 1 1 1
1 1 1 1

Answer 1

The answer is the number of combinations of n-1 out of (n+m-2), or how many ways to choose n-1 objects out of n+m-2.
This is because it is necessary to take n-1 steps down and m-1 steps to the right. The only question is in what order to do them. In total, n-1+m-1 steps will be taken, and from them you need to choose some n-1 steps down, the rest will be steps to the right. This is where combinations come in.
It can be considered Pascal's triangle, it will turn out exactly the same as described by poznavaka ,
it can be calculated using the factorial formula: (n + m-2)! /(n-1)! /(m-1)!
For your example where N=3 and M=4 the answer would be 5!/2!/3! = 120/2/6 = 10.

Answer 2

Not at all. It is impossible to come to the lower right corner.

Answer 3

It's simple: 0 possible outcomes. As stated in the condition - from the upper left corner it is impossible to reach the lower right corner, since only down or left movement is available.

Answer 4

The answer is 10.