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 the first letter for the initials in 26 different ways, as there are 26 letters in the English alphabet.

For each of the 26 ways we can choose the first initial, we have 25 different ways we can choose the second letter. It is only 25 because the letters cannot be repeated in the three-letter initials. In this way, we won’t repeat the first letter.

For each of the 26 ways we can choose the first initial, we have 25 ways we can choose the second letter. And, we also have 24 different ways of choosing the third initial. By choosing only from 25 letters, we make sure that we are not repeating the first and second initials.

From the previous description, it follows we must use the product rule.

__Answer:__

By the product rule, people can have 26x25x24=15600 different three-letter initials with no letter repeated.

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