Q. 1.22 - A First Course in Probability

Question

In Problem 21, how many different paths are there from A to B that go through the point circled in the following lattice?

Step-by-Step Solution

Answer

The total number of paths from A to B is $18$.

Step 1. Given information.

To go from point A to B, a person has to go one step up or one step to the right at each move. To reach point B through the circled point, there are two paths - One is from A to the circled point and the other is from circled point to B.

Step 2. Find the number of paths.

Let the circled point be C

No. of ways to reach from A to C is $2$, which are right then up then right then up or up then right then up then right.

So, number of paths from A to C $=\frac{4!}{2!2!}=\frac{4\times 3\times 2!}{2\times 1\times 2!}=6$

No. of paths from C to B = $3$ ($2$ steps right and $1$ step up)

Therefore, total number of paths from A to B = $6\times 3=18$.