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 can choose the first initial in 1 way. That is, the three-letter initial must begin with an A.
Then, we can choose the second initial in 26 different ways. Notice that the English alphabet has 26 letters.
For each of the 26 ways we can choose the second initial, there are 26 ways we can choose the third one.
Answer:
By the product rule, there are 26×26=676 different three-letter initials that start with an A.
Notice that before I give the final answer, I describe the solution of the problem in the same way the principles are stated. That makes it easy for us to decide what principle we can apply in the solution to the exercise.
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?
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- 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?
- 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?
