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Take into consideration a 10\times 9 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

92376


Option: 2

93375


Option: 3

93378


Option: 4

92378


Answers (1)

best_answer

We must take 10 steps to the right and 9 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 10 R steps and 9 U steps in total.

These 10 Rs and 9 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} & { }^{19} \mathrm{C}_9=\frac{19 !}{9 !(19-9) !} \\ = & \frac{19 !}{9 ! \times 10 !} \\ = & 92378 \end{aligned}
Total number of ways : 92378 . 

Posted by

Gautam harsolia

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