Question

From Goa to Bombay there are two routes, air and sea. From Bombay to Delhi there are three routes; air, rail and road. From Goa to Delhi via Bombay, how many kinds of routes are there?

Answer 1

Hint: The fundamental counting principle states that if one event $ A $ can be done in $ m $ ways and the second event $ B $ can be done in $ n $ ways then both events together can be done in $ m \times n $ ways. We will use this principle to solve the given problem.

Complete step-by-step answer:
Let us consider the event $ A $ and event $ B $ as following:
Event $ A $ : The person is travelling from Goa to Bombay.
Event $ B $ : The person is travelling from Bombay to Delhi.
In this problem, it is given that there are two routes from city Goa to city Bombay. Therefore, we can say that events $ A $ can be done in $ 2 $ ways. Let us say $ m = 2 $ .
Also it is given that there are three routes from city Bombay to city Delhi. Therefore, we can say that event $ B $ can be done in $ 3 $ ways. Let us say $ n = 3 $ .
Now we need to find the total number of possible routes from Goa to Delhi, via Bombay. That is, we need to find the total number of possible routes when a person is travelling from Goa to Bombay and then Bombay to Delhi. That is, we need to find the total number of possible routes when both events $ A $ and $ B $ happen. For this, we will use the fundamental counting principle. The fundamental counting principle says that if one event $ A $ can be done in $ m $ ways and the second event $ B $ can be done in $ n $ ways then both events together can be done in $ m \times n $ ways. Therefore, in this problem both events $ A $ and $ B $ can be done in $ m \times n = 2 \times 3 = 6 $ ways. 

Therefore, we can say that there are a total $ 6 $ different (distinct) possible routes from Goa to Delhi, via Bombay.

Note: To solve this problem, we can think in a different way. That is, to travel from Goa to Bombay, a person can select one route out of two routes in $ {}^2{C_1} = 2 $ ways. Similarly, to travel from Bombay to Delhi, a person can select one route out of three routes in $ {}^3{C_1} = 3 $ ways. Therefore, to travel from Goa to Delhi via Bombay, there are total $ {}^2{C_1} \times {}^3{C_1} = 2 \times 3 = 6 $ different (distinct) ways.