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When buying groceries earlier today, the self-serve checkout asked me:
Combien de sacs du magasin avez-vous utilisés aujourd'hui?
The "du" surprised me. The self-serve checkout, when its language is set to English, asks me "How many store-provided bags did you use today?", so I suppose "sacs du magasin" means "the store's bags".
I'm wondering if "sacs de magasin" makes sense, to mean "store bags"?
Yes, this person was (arguably) implicitly refering to this very store's bags. It's quite safe to assume he/she cares far less about the number of other bags you used, since you couldn't be charged for them (hopefully!)...
There are a number of terms to refer to these bags, varying between different areas of the world, or regions in France, and it also depends on the specifics (plastic ones, paper ones, ...).
s“Sac de magasin” would be grammatically correct French, but it would mean a bag which is generically associated with stores (it could be a kind of bag that is made by stores, or a kind of bag for use in stores, or some other association, similar to “store bag” in English). The expression “sac de magasin” is not commonly used for anything in French. If you brought your own bags to use in shops, they could be called “sacs de magasin” if we ever used this expression in French. Grammatically, here, “de” is the preposition de (“of”) plus a partitive article.
“Sac du magasin” means a bag of “the” store. In context, “the” refers to the store that you're in. In this context, it clearly means a bag provided by the store. The association between the bags and the store could be different, for example it could mean a bag decorated with the store's brand, although in practice those would be provided by the store anyway. Grammatically, “du” is the preposition de (“of”) plus a definite article (in masculine singular form).
