"Don't you have a swimsuit?"
Translation:אין לךְ בגד ים?
In this context, the word "רשות" means "domain". The suffix "כן-" is for second-person plural feminine. Therefore, "אין ברשותכן בגד ים?" literally means "Is not there a swimsuit in your domain?" but it doesn't sound strange in Hebrew.
So the difference between this and "אין לך" would be similar to english "Don't you have...?" and "Don't you own...?" the latter being slightly more formal form of the former? (formal form of the former, all hail alliterations!)
I'm not a native english speaker, but may be "don't you have a swimsuit at your disposal?" is more appropriate and closer to origin?
In the multiple choice I didn't have the option of אין לךְ בגד ים. I only had the option which included the word ברשותכ. I knew the other two options were not correct, so looked up mean of the word in Google Translate. Turns out I got it correct, but was never taught that word in previous lessons.
bir-shut-khen - בִּרְשׁוּתְכֶן - second person, plural, fem.
bir-shut-kha - בִּרְשׁוּתךָ - second person, singular, masc.
Don't actually know, but Google Translate shows it having these vowels in the singular form: בִּרְשׁוּתךָ
I looked them up in Reverso and it looks like שלך is basically "your" and ברשותך would be more like "you own" or "you possess." I'm not a native speaker, so not sure if I'm totally correct.