These questions are about calculating the number of unique paths from one corner of a grid to the opposite corner.

Let us take an example where you have to move from point A to point B in the given grid. You can move only along the positive X-axis or the positive Y-axis.

To calculate the number of different paths, first note that you will always have to take 7 steps, out of which 4 are along the x-axis and the remaining 3 are along the y-axis. Let us call the steps along the x-axis “x” steps and the steps along the y-axis “y” steps.

Next, note that interchanging any x step with a y step will result in a different path, while interchanging any two x steps or any two y steps will result in the same path. So the question finally reduces to finding the number of ways of arranging 4 x's and 3 y's in a line:

x x x x y y y

This can be done in \(\dfrac{7!}{4! \times 3!}\) ways, as there are 4 identical x's and 3 identical y's.

To understand this better, let us take three different arrangements of x's and y's. Each arrangement results in a unique path. Here, we have shown three paths corresponding to the three arrangements: xxxxyyy, xxyyxxy, and xyxyxyx.

In general, in a grid of size \(m \times n\), the total number of paths is

\[\bbox[5px, border: 2px solid #0071dc]{^{m+n}\mathrm{C}_m \;\text{or}\; ^{m+n}\mathrm{C}_n}\]