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Think about a 14\times 13 grid with A as the starting point and B 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 B .

Option: 1

20058300


Option: 2

2058300 


Option: 3

2005300


Option: 4

205830


Answers (1)

best_answer

We must take 14 steps to the right and 13 steps up in order to go for the shortest distance to endpoint B. Let R and Ueach represent a step to the right and an upward step, respectively. Consequently, there are 14 R steps and 13 U steps in total.

These 14 Rs and 13 Us can be arranged in any way to find the shortest path. The binomial coefficient provides the number of possible arrangements for these steps:
{ }^n \mathrm{C}_r=\frac{n !}{r !(n-r) !}
where n denotes the overall number of steps, r the number of steps taken in one particular direction, in this case to the right, and \left ( n-r \right ) the number of steps taken in the opposite direction, in this case upwards.

Using this equation, we obtain:

\begin{aligned} & { }^{27} \mathrm{C}_{13}=\frac{27 !}{13 !(27-13) !} \\ = & \frac{27 !}{13 ! \times 14 !} \\ = & 20058300 \end{aligned}
Total number of ways : 20058300 . 

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Gaurav

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