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Think about a 13\times 14 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

2005800


Option: 4

205830


Answers (1)

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We must take 13 steps to the right and 14 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 13 R steps and 14 U steps in total.

These 13 Rs and 14 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}_{14}=\frac{27 !}{14 !(27-14) !} \\ = & \frac{27 !}{14 ! \times 13 !} \\ = & 20058300 \end{aligned}
Total number of ways : 20058300 . 

Posted by

Gautam harsolia

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