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Think about a 11\times 12 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

135078


Option: 2

152078


Option: 3

1352078


Option: 4

13578


Answers (1)

best_answer

We must take 11 steps to the right and 12 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 11 R steps and 12 U steps in total.

These 11 Rs and 12 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} & { }^{23} \mathrm{C}_{12}=\frac{23 !}{12 !(23-12) !} \\ = & \frac{23 !}{12 ! \times 11 !} \\ = & 1352078 \end{aligned}
Total number of ways : 1352078 . 

Posted by

Sanket Gandhi

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