Take into consideration a grid with as the starting point and as the finishing point. Each step can either be an upward or a rightward movement. Find the total number of shortest paths that lead to endpoint .
We must take steps to the right and steps up in order to go for the shortest distance to endpoint . Let and each represent a step to the right and an upward step, respectively. Consequently, there are steps and steps in total.
These Rs and Us can be arranged in any way to find the shortest path. The binomial coefficient provides the number of possible arrangements for these steps:
where n denotes the overall number of steps, the number of steps taken in one particular direction, in this case to the right, and the number of steps taken in the opposite direction, in this case upwards.
Using this equation, we obtain:
Total number of ways : .
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