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Take into consideration a 8\times 7 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

6425


Option: 2

6435


Option: 3

6135


Option: 4

6465


Answers (1)

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We must take 8 steps to the right and 7 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 8 R steps and 7 U steps in total.

These 8 Rs and 7 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} & { }^{15} \mathrm{C}_7=\frac{15 !}{7 !(15-7) !} \\ = & \frac{15 !}{7 ! \times 8 !} \\ = & 6435 \end{aligned}
Total number of ways : 6436 . 

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rishi.raj

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