Yup. That's basically it although I think their question was mainly on the notation but you covered it. This is a case where we do need to care if Y is continuous or not though (which you mentioned but it deserves an extra special call out here)
It's still not clear to me what you're actually trying to calculate the standard deviation of. It sounds like you're not entirely clear on that either. If you don't know what you want it's very hard for me to do anything.
What are you looking to find the standard deviation of exactly? Do you want to know what the standard deviation would be if you literally ignored the grouping, considered all the data as a single dataset and then calculated the standard deviation? Or are you interested in what the standard...
All of your work for that is correct except for one thing. In both cases where your calculating using the "choose" function you're getting it wrong.
4C4 is 1 and 4C3 is 4. I'm not sure how you're getting your answers of 1/6 and 2 but that seems to be your issue with that problem.
It was pretty obvious that hlsmith was asking you to explain your terminology. What is DEA? Acronyms aren't universal and can mean different things in different contexts.
Also please share what you've tried and why they didn't work.