"Ladies and gentlemen!"
Translation:Panie i panowie!
Thanks for this clear answer. I take you to be saying that in theory vocative would be used. My assumption is that with Pan and Pani vocative is the same as nominative.
Nominative mixed plural would be “Państwo”. But I think you are also saying (in your comment above 4 years ago) that no matter what the strict theory may be the practice is to separate the genders and use „Panie i panowie” or „Proszę państwa”.
Not clear how the genitive “państwa” gets in here. But, in summary, I guess this is simply a further example of an exception to the strict rule.
The vocative of "Pan" is "Panie", but for "Pani" it's still "Pani". The rule of vocative being the same as nominative only works in plural. In singular it can be different, but this course doesn't cover that case at all.
There are a couple ways of addressing a crowd formally, some of which were mentioned in other comments. But simply saying "Państwo" (in vocative) is just awkward.
"Proszę państwa" is a verb phrase, consisting of the verb "prosić" (in first person singular) and "państwo" in a case that matches the verb. Honestly, I'm not 100% sure if it's genitive or accusative (probably the latter), because this verb in that meaning is not really used with inanimate nouns.
I think it is strangely Genitive, although literal "I am asking X (for something)" would take Accusative.
But when this phrase is used as a kinda-Vocative, it's Genitive, e.g. "Proszę szanownej pani". Actually even just "Proszę pani" shows it, as Accusative would be "Proszę panią".