There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?

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, let’s describe the solution to the problem in a similar way to the principles above.

We can choose four major auto routes from Boston to Detroit. For each of the previous four major routes, we can choose six other routes from Detroit to Los Angeles.

Therefore, by the product rule, there are 4×6=24 major auto routes from Boston to Los Angeles via Detroit.

Notice how the description we did is similar to the product rule and different from the sum rule. That helps us to decide which rule we must apply.

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