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  • Think about a  grid with <img alt="A" src="https://entrancecorner.oncodecogs.com/gif.l

Think about a 5\times 4 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

32


Option: 2

64


Option: 3

126


Option: 4

128


Answers (1)

best_answer

We must take 5 steps to the right and 4 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 5 x steps and 4 y steps in total.

These 5xs and 4 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} & { }^9 \mathrm{C}_4=\frac{9 !}{4 !(9-4) !} \\ = & \frac{9 !}{4 ! \times 5 !} \\ = & 126 \end{aligned}
Total number of ways : 126 . 

Posted by

Rishabh

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