Question

Suppose there are 7 roads connecting town a to town b and 6 roads connecting town b to town

Answer 1

First, we should answer two simple questions.

1. How many ways can we travel from a-b?

2. How many ways can we travel from b-c?

This is given in the problem - because there are 7 roads connecting a to b, there are 7 ways to get from a-b.  Because there are 6 roads from b-c, there are 6 ways to get from b-c.

Now that we understand this, we can use some logic to figure out the rest of the problem.  Let's think about each case.

Let's go from a-b.  We'll choose road 1 of 7.  Now that we are in b, we have 6 more choices.  This means that there are 6 ways to get to from a-c if we take road 1 when we go to b.

If we take any road going from a-b, there will be 6 options to get from b-c.

So, we can just add up the number of options because we know that there are 6 routes per road from a-b.  This is simply 7*6 = 42.  So, there are 42 ways to travel from a to c via b.