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Take into consideration a 3\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

14


Option: 2

21


Option: 3

35


Option: 4

70


Answers (1)

best_answer

We must take 3 steps to the right and 4 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 4 U steps in total.

These 3 Rs and 4 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} & { }^7 \mathrm{C}_4=\frac{7 !}{4 !(7-4) !} \\ = & \frac{7 !}{4 ! \times 3 !} \\ = & 35 \end{aligned}

Total number of ways : 35

Posted by

rishi.raj

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