Question: You are given 20 sugar bars (B1,B2,..B20) and 50 salt bars (S1, S2..S50). Consider subsets of these 70 bars, consisting of 30 bars, exactly 12 of which are sugar bars. How many such subsets are there? (Answer: 2.2742 * 10^18)
Attempt: The consisting of 30 bars part confuses me. Like, am I supposed to find the subset of a subset here? From 70 total bars, a subset of 30 bars where there are 12 sugar bars and therefore 18 salt bars exist. So, I'm assuming I use combinations here? Am I finding the number of subsets that consist of 30 bars with 12 being sugar bars and a different combination of salt bars?
What if the question was asking You are given 20 sugar bars (B1,B2,..B20) and 50 salt bars (S1, S2..S50). Consider subsets of these 70 bars, that contain at least 3 of which are sugar bars (and any number of salt bars). How many such subsets are there?
What would the difference in approach be when the wording changes from "exactly" to "at least"
you're working with 70 bars, 50 salty and 20 sugary. We only care about the subsets that have 12 sugary, it could be any 12 from the total 20 that are sugary so we have a combination ${20}\choose{12}$ that lets us know all the possible combinations of 12 bars that we can work with. Since we're only interested in the subsets with 30 bars in total we have to have 18 salty bars. But similarly we want all possible combinations of the 18 from 50, so we end up with ${50}\choose{18}$. The product ${20}\choose{12}$${50}\choose{18}$ gives us the answer. If we were to say that the subsets have to have at "least" 12, we would have to add the products of the different combinations , ${20}\choose{12}$${50}\choose{18}$+${20}\choose{13}$${50}\choose{17}$+ $\cdots$ +${20}\choose{20}$${50}\choose{10}$