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Think about a 3\times 2 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

10


Option: 2

20


Option: 3

30


Option: 4

40


Answers (1)

best_answer

We must take 3 steps to the right and 2 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 3 R steps and 2 U steps in total.

These 3 Rs and 2 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} & { }^5 \mathrm{C}_2=\frac{5 !}{2 !(5-2) !} \\ = & \frac{5 !}{2 ! \times 3 !} \\ = & 10 \end{aligned}
Total number of ways : 10 . 

Posted by

Divya Prakash Singh

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