I have a few different types of counting problems where I feel I don't understand the approach to specific counting questions. I will list down the different scenarios with how I logically see it.
Scenario 1
For $Q1$:
To find the total number of subsets would be $70\choose30$.
Exactly 12 Beer, so exactly 18 Cider for a 30 bottle subset.
So, total count: $20\choose12$ $50\choose18$
Ok, that was simple. However, what if the questions asked me:
1) 30 bottle subsets where there were at least 3 beer bottles? How many such subsets exist?
My thinking is whenever I see the words "at least x", I have to look at what is below x. So, in this case it's 3. I will then look at beer bottles for 0, 1, 2.
$20\choose0$ $50\choose30$ + $20\choose1$ $50\choose29$ + $20\choose2$ $50\choose28$. Is this correct way to look at it?
Now, what if it was:
2) 30 bottle subsets where there were at most 3 beer bottles? How many such subsets exist?
Do I look at subsets for beer bottles from 4-20 for this case? That would take me forever to do if I approached it like how I did for the previous case.
3) 30 bottle subsets where there were at least 3 beer bottles and/or at most 12 cider? How many such subsets exist?
In this case, I have a limit now. How would I approach this for the case of and & the case for or?
Also, for Question 2)
Does any number of beer/cider bottles mean I have to count it in the form of $2^x$. Or is it $choices^{number of bottles}$
I am confused on when to use the choose function, when to use the exponential function in these counting scenarios.
Like for 2, the answer is option d. Do we subtract $20\choose12$ $50\choose12$ because they are double counted? How would I ever know whats double counted?
Like for this question, the answer is c) but I don't get the subtracted part.
Scenario 2
For these password questions, again when I see at least and at most what should I be doing?
Answer: (Q1-b, Q2)-c)
When should I use exponentials to count and choose functions to count? I need some clarification on how to break down these problems. I've been struggling to conceptualize it so far from my attempts.