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Think about a 8\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

23102


Option: 2

24410


Option: 3

43100


Option: 4

24310


Answers (1)

best_answer

We must take 8 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 8 R steps and 9 U steps in total.

These 8 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} & { }^{17} \mathrm{C}_9=\frac{17 !}{9 !(17-9) !} \\ = & \frac{17 !}{9 ! \times 8 !} \\ = & 24310 \end{aligned}

Total number of ways : 24310 . 

Posted by

Suraj Bhandari

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