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Take into consideration a 9\times 10 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

92378


Option: 2

93378


Option: 3

92478


Option: 4

92578


Answers (1)

best_answer

We must take 9 steps to the right and 10 steps up in order to go for the shortest distance to endpoint B . Let x and y each represent a step to the right and an upward step, respectively. Consequently, there are 9x steps and 10y steps in total.

These 9xs and 10ys 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}_{10}=\frac{19 !}{10 !(19-10) !} \\ = & \frac{19 !}{10 ! \times 9 !} \\ = & 92378 \end{aligned}
Total number of ways : 92378 . 

Posted by

Rishabh

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