I am not really understanding how to solve this set of questions which are involving premutations and combinations, can someone explain question (a) or more?
4.4 How many ways are there to choose a bag of 6 donuts from 8 varieties,
(a) if there are no two donuts in the bag of the same variety?
(b) if all donuts in the bag are of the same variety?
(c) if there are no restrictions on the contents of the bag?
(d) if there must be at least two varieties in the bag?
(e) if there must be at least three blueberry-filled donuts in the bag?
(f) if there can be no more than two blueberry-filled donuts in the bag?
Presumably, we do not care in what order the donuts are in our bag. They will all go home and get eaten either way.
In part (a), we want to choose six flavors from eight total. We will then get one donut of each of those flavors.
In part (b), we simply want to choose one flavor from 8 total. We will then get eight donuts of that one flavor.
In part (c), apply stars-and-bars. You can imagine setting up a sequence of $\star$'s and $|$'s, with $6$ stars(the number of donuts we will buy) and $7$ bars (one less than the number of flavors available). Get as many of the first flavor as there are stars to the left of the first bar. Get as many of the second flavor as there are stars inbetween the first and second bar. Etc...
In part (d), apply inclusion-exclusion. If the answer in part (c) is with no restriction, and the "bad" options were those options we found in part (b), how many are "good" options?
In part (e), go ahead and put three blueberry-filled donuts into your bag before anything else. You still need to get three more donuts with no restriction. Use a similar technique as part (c).
In part (f), apply inclusion-exclusion. If the answer in part (c) is with no restriction, and the "bad" options were those options we found in part (e), how many are "good" options?