There are a lot of math questions here. I'll answer the first 5 questions to get you started.
=============================
Problem 19
This is a permutation because order matters and we want to find out the number of ways to arrange the people in the first, second, third and fourth place slots.
You would use the nPr formula.
=============================
Problem 20
We use the counting principle here. This is because we can create a tree diagram. To find the number of branches in this diagram, you would multiply out all the values.
==============================
Problem 21
This is similar to problem 19; however, order doesnt matter in this case. All that matters is the grouping of books. For example, the group {A,B,C} is the same as {B,C,A} because both groups have the same books in them and the the groups have the same number of books. So in short, we use a combination here instead of a permutation.
You would use the nCr formula.
==============================
Problem 22
Similar to problem 21. Order does not matter so we use a combination. There is no natural way to order or rank the patients being tested, and there's no need to do so, because all that matters is the group itself rather than specific individuals.
==============================
Problem 23
Again, order doesn't matter because we arent worried about rank or position. All we care about is the group. We use a combination here as well.
