Q. 1.21 - A First Course in Probability

Question

Consider the grid of points shown at the top of the next column. Suppose that, starting at the point labelled A, you can go one step up or one step to the right at each move. This procedure is continued until the point labelled B is reached. How many different paths from A to B are possible? Hint: Note that to reach B from A, you must take $4$ steps to the right and $3$ steps upward.

Step-by-Step Solution

Answer

The possible number of paths from A to B is $35$.

Step 1. Given information.

It is given that to reach B from A, $4$ steps are to be taken to the right and $3$ steps up. So, total $7$ steps are to be taken.

Step 2. Find the possible number of paths.

Using multinomial rule, the possible number of paths from A to B is given by $\frac{m!}{n_1!n_2!}$

where,

$m=$Total number of steps $=7$

$n_{1=}$Total number of steps taken to the right $=4$

$n_2=$Total number of upward steps $=3$

Therefore, the possible number of paths from A to B $=\frac{7!}{4!3!}=\frac{7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!}=35$