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.
Related exercises:
- How many bit strings are there of length eight?
- How many different three-letter initials with none of the letters repeated can people have?
- A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?
- How many different three-letter initials are there that begin with an A?
- A multiple-choice test contains 10 questions. There are four possible answers for each question
- Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco?