Need Help w/ Permutations

There is a question in my math book that asks "How many permutations are there on the symbols: A, B, C, D, E and F taken 3 at a time?"
According to my counting, that is a total of 6 symbols. How my math book teaches this is that there are three symbols being taken at a time, so there are three positions the symbols could allocate. I believe it should then be solved as such:
Three positions (taken three at a time): _ _ _
First position can be any of the 6 symbols: 6 _ _
Second position can be taken by any of the remaining 5 symbols: 6 x 5 _
Third position can be taken by any of the remaining 4 symbols: 6 x 5 x 4 = 120.
Evidently, the answer my math book gives is 60 for this question, which would make since with work such as: 5 x 4 x 3, which is similar to what I did, however it starts with 5 instead of 6, when there are clearly 6 symbols (unless of course I am blind).
So I'm wondering why exactly it is that this is how I am supposed to answer the question, or if I'm looking at it wrong and there's an entirely different way to answer this question. Looking back at previous examples in the book, the work I assume is done (5 x 4 x 3) is not similar to any of them, but it is in fact the answer, so can someone please explain to me how/why it is the answer?
Thanks for any help anyone can offer me!

teetar said:

"How many permutations are there on the symbols: A, B, C, D, E and F taken 3 at a time?"

I believe it should then be solved as such:
Three positions (taken three at a time): _ _ _
First position can be any of the 6 symbols: 6 _ _
Second position can be taken by any of the remaining 5 symbols: 6 x 5 _
Third position can be taken by any of the remaining 4 symbols: 6 x 5 x 4 = 120.

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I agree with both your method and your answer. (Thank you, by the way, for showing your work so nicely!) This online lesson, in the section on "permutations", gets the same answer, too.

I think the book is wrong.

stapel said:

I think the book is wrong.

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I thought this as well (in fact, a similar problem occurred with another problem in the book) however, my math teacher has told me that the book's answers are in fact correct, so I'm racking my brain trying to figure out how, but I can't seem to find any answer.

Thank you for your input on this question!