Let’s review the sum rule.
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.” Discrete Mathematics and its applications by Rosen.
It is easy to see in this case that none of the 37 offices on the first floor is the same as the 37 offices on the second floor. Also, the same applies to the rest of the floors.
So, you can apply the sum rule by adding all the offices from each floor. You sum 37 offices 27 times, which is the same as 27×37 = 999 offices.
Another way of solving this type of problem is to modify the question into a similar one, that answers the same problem.
For instance, we can modify the question as follows:
An office building contains 27 floors and has 37 offices on each floor. In how many ways we can choose 1 office?
As you can see, with this question we are answering how many offices are in the building.
So, if you choose 1 of the 27 floors, for each floor you can choose 1 of the 37 offices, it is clear that you can apply the product rule. See the definition of the product rule below.
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.” Discrete Mathematics and its applications by Rosen.
So, you can answer that there are 999 (27×37) offices by using the product rule.
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