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  • Take into consideration a grid with <img alt="A" src="https://entrancecorner.oncodecogs.com/gif.latex?A"

Take into consideration a 6\times 5 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

461


Option: 2

462


Option: 3

463


Option: 4

464


Answers (1)

best_answer

We must take 6 steps to the right and 5 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 6 R steps and 5 U steps in total.

These 6 Rs and 5 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} & { }^{11} \mathrm{C}_5=\frac{11 !}{5 !(11-5) !} \\ = & \frac{11 !}{5 ! \times 6 !} \\ = & 462 \end{aligned}

Total number of ways : 462

Posted by

vishal kumar

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