Question

There are 4 different routes between cities A and B, and 3 different routes between cities B and C. How many different routes are there from city A to city C by way of city B?

Answer 1

Obviously, one can go from city A to city B by any of the 4 routes, i.e., in 4 ways. After having gone to B by any of the different 4 routes, one can go to city C by any of the three routes. 

Thus, corresponding to one route taken, from A to B, he has 3 choices from B to C. Therefore, corresponding to 4 routes, there are 12 choices in all.

Therefore, he can go from A to C via B, in 4 × 3 = 12 ways as depicted in the given tree diagram.

The possible routes taken are :

Total number of ways =12. 

Thus, the examples discussed above illustrate the use of a general principle, called the product rule or the fundamental principle of counting, which is stated below.