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Think about a6\times 7 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

1714


Option: 2

1716


Option: 3

1718


Option: 4

1720


Answers (1)

best_answer

We must take 6 steps to the right and 7 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 7 U steps in total.

These 6 Rs and 7 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 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} & { }^{13} \mathrm{C}_7=\frac{13 !}{7 !(13-7) !} \\ = & \frac{13 !}{7 ! \times 6 !} \\ = & 1716 \end{aligned}
Total number of ways : 1716 . 

Posted by

shivangi.shekhar

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