Relevant definitions for this exercise:

THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are n1n2 ways to do the procedure.”

THE SUM RULE: “If a task can be done either in one of n1 ways or in one of n2 ways, where none of the set of n1 ways is the same as any of the set of n2 ways, then there are n1 +n2 ways to do the task.”

The definitions were taken from the textbook Discrete Mathematics and its Applications by Rosen.

Now we can solve the exercise.

We want to go from New York to San Francisco via Denver picking an airline for the flight to Denver, and an airline for the flight from Denver to San Francisco.

We can choose an airline to travel from New York to Denver in 6 different ways. The problem description states that 6 different airlines flights on that route.

For each of the 6 different ways that we choose to fly from New York to Denver, there are 7 ways we can fly from Denver to San Francisco (as per the exercise description).

**Answer:**

Therefore, by the product rule, there are 42 different pairs of airlines that we can choose to book a trip from New York to San Francisco via Denver when we pick 1 airline to go to Denver, and another one to go to San Francisco.

Notice that we should always give a final answer to the exercise. Sometimes is not enough just to write a number as an answer, we need to elaborate an answer according to the question we were asked.

Notice the final answer to this exercise and the question we were asked. The rest is what we did to give the final answer.

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