How can I approach the problem below? How can I figure out what to do? All I know is that it doesn't want two books to be of the same language, so the answer is less than 5 * 6 * 8.

Can someone please explain how I can make this problem work with the Addition and Multiplication principle?

There are five different Spanish books, six different French books, and eight different
Transylvanian books. How many ways are there to pick an (unordered) pair of two
books not both in the same language?

For each of the 5 Spanish books, the other book in the pair can be any of the French or Transylvanian books. That is, there are 5×(6+8) possible pairs that include a Spanish book. Then there are 6×8 possible pairs that include no Spanish book (French and Transylvanian only).

The total count of unordered pairs is ...

5×(6+8) + 6×8 = 70 +48 = 118

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This is effectively the total of all the pairwise products of the numbers of books.

5×6 + 5×8 + 6×8 = 118

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If you consider an extended version of this problem, where there are different numbers of books in 5 or 50 languages. You see that you have to consider each possible pair of products.

If you had a 3-book set instead of a 2-book set in perhaps a dozen languages, you would need to consider all the ways you could choose 3 from a dozen, then figure the products of numbers of books for each of those ways. (Tedious, but not complicated.)