A, B and C are three cities. There are 5 routes from A to B and 3 routes from
Question:

A, B and C are three cities. There are 5 routes from A to B and 3 routes from B to C. How many different routes are there from A to C via B?

Solution:

Given: 5 routes from A to B and 3 routes from B to C.

To find: number of different routes from $A$ to $C$ via $B$.

Let $E_{1}$ be the event : 5 routes from A to B

Let $E_{2}$ be the event : 3 routes from $B$ to $C$

Since going from $A$ to $C$ via $B$ is only possible if both the events $E_{1}$ and $E_{2}$ occur simultaneously.

So there are $5 \times 3=15$ different routes from $A$ to $C$ via $B$.