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Take into consideration a 4\times 3 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

7


Option: 2

14


Option: 3

28


Option: 4

35


Answers (1)

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

These 4 Rs and 3 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}_3=\frac{7 !}{3 !(7-3) !} \\ = & \frac{7 !}{3 ! \times 4 !} \\ = & 35 \end{aligned}
Total number of ways : 35 . 

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