The full sentence: Ils ont besoin de livres et de DVDs pour pouvoir s'occuper pendant le voyage.
My understanding is that livres et DVDs are plural, so the contracted form of de + les would be des. So why is "de" correct? I don't see a negation??
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It can be hard for us English speakers to know when to use des and when to omit it, since we don't really have it in English. For us, "books" sounds like it has the same status in these sentences:
There are books on the table.
I have a lot of books.
I need books.
Yet French sees them differently:
Il y a des livres sur la table.
J'ai beaucoup de livres.
J'ai besoin de livres.
The first case tends to be a hurdle, particularly if the student has already learned that des is de + les. Why should there be anything definite in that sentence?
But interestingly enough, in the other examples, things are easier. The rule is that after a container or quantity, i.e. when you have an expression that already uses de, you leave out the article.
Beaucoup de livres
Besoin de livres
Peu de livres
Plus de livres
Plein de livres
And then if you add the article (les) back in, the meaning changes in a very predictable way: "of the" ("many of the books", "need the books", "few of the books").
So the real question is why the article is present in the standard case! I don't have a satisfactory answer for that on hand... but the good news is that the rule is at least mostly consistent: After a container or quantity, de is enough on its own for an indefinite noun, whether singular or plural.
Ils ont besoin de livres et de DVDs = they need books and DVD (can be any books or DBDs) Ils ont besoin DES livres et DES DVDs = they need THESE books and THESE DVD (implying the books were mentionned before, ie they have a list of books to take for traveling: Do they need to bring the books AND the DVDs? Oui , ils ont besoin DES livres et DES DVDs.