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Take into consideration a 15\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

775760


Option: 2

7758760


Option: 3

77558760


Option: 4

775860


Answers (1)

best_answer

We must take 15 steps to the right and 14 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 15 x steps and 14 y steps in total.

These 15 xs and 14 ys 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} & { }^{29} \mathrm{C}_{14}=\frac{29 !}{14 !(29-14) !} \\ = & \frac{29 !}{14 ! \times 15 !} \\ = & 77558760 \end{aligned}
Total number of ways : 77558760 .

Posted by

Pankaj

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