Learn Polish: Na państwa życzenie kawa jest bezpłatna. - At your request, the coffee is free of charge.

In this sentence, państwa acts as the formal plural possessive "your" (when addressing a group of people). Grammatically, it is the genitive case of the noun państwo (ladies and gentlemen). So literally, the phrase na państwa życzenie translates to "at the request of ladies and gentlemen".

Why is państwa lowercase here? Aren't formal words for "you" usually capitalized?

In direct written correspondence, like an email or a printed sign addressing a customer, you will almost always see it capitalized as Państwa. In general texts, grammar examples, or transcripts of spoken Polish, it is often kept lowercase.

What case is życzenie in the phrase na państwa życzenie?

It is in the accusative case. The preposition na combined with the accusative is used for fixed phrases about responding to an event or action, such as na życzenie (at the request) or na zaproszenie (at the invitation). Because życzenie is a neuter noun, its accusative form is identical to its dictionary form.

Can I use darmowa instead of bezpłatna to mean "free"?

Yes, they are synonyms! However, bezpłatna (literally "without payment") sounds slightly more formal and professional. You will often see it on official signs, in hotels, or in restaurants, which perfectly fits the polite tone of using państwa.