What is the essential formula for a grid pathing problem with equal moves Up and Right?

In grid pathing problems where you need to make equal moves in two directions—Up and Right—you're essentially trying to determine the number of unique paths from one corner of a grid to another by taking a set number of Up and Right moves.

The essential formula involves calculating the total permutations of the moves. If you have a total of n moves to make, where n1 is the number of Up moves and n2 is the number of Right moves, the total number of unique paths can be calculated using the factorial function. The formula captures the arrangement of these moves as permutations of a multiset, which takes into account the indistinguishable nature of the same type of moves (i.e., all Up moves are identical, and all Right moves are identical).

The correct formula is derived as follows: the total arrangements of n1 Up moves and n2 Right moves can be given by the expression (n1 + n2)! divided by the product of the factorials of the individual counts of the moves, n1! for Up moves and n2! for Right moves. This simplifies the counting process by eliminating the redundancies introduced by the identical moves.

Therefore, the formula (n1 + n2)! / (n1!