Let’s refresh the rules we can use for this type of 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 previous definitions are in the textbook: Discrete Mathematics and its applications by Rosen.
The easier way to solve this type of exercise for me is to express the problem as one of the rules definition. That’s why I always start by having the definitions clear.
We can choose 26 letters (the number of letters in the English alphabet) as the first initial.
In how many ways can we choose the second initial? You are right, in 26 ways.
And the same goes for the third one.
Notice that you can choose the second letter without regard to the letter you choose as the first one. Also, the same applies to the third one.
So, we can do the first task (choosing the first initial) in 26 ways. For each of the 26 ways we can do the first task, there are 26 ways of doing the second one. Lastly, for each of the 26 ways, and each of the 26 ways we can do the second one, there are 26 ways that we can do the third one.
After this description of the problem, we can be sure that we need to apply the product rule.
Answer:
People can have 17576 (26x26x26=263) different three-letter initials.
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