A question was recently asked on our ZIML discussion forum about these questions that I thought would better fit here.

Here's some hints/thoughts on these MC II-A Questions. All of these are good practice for sets and inclusion-exclusion.

4.27: Use inclusion-exclusion with A = "6-digit numbers with 1 next to 0" and B = "6-digit numbers with 1 next to 2". Note we can then use the grouping trick to help. Caution: When dealing with 0, we need to ensure it's not the first digit of the number. For example, when counting the number of elements in A, it might help to do one case where we have 01 (so this cannot go first) and another case for 10. Example 4.7(b) from the textbook can be a good problem to review here to help as well.

4.28: Don't overthink this one. You have enough information to start by finding how many red balls there are in the bag.

4.30: Part (b) shows how this type of question can get a little messy if you're not using inclusion-exclusion. In (b), there are two cases to consider (i) 2 friends get 1 book and the last gets 3 books, or (ii) 2 friends get 2 books and the last gets 1 book. For (a) use complementary counting and inclusion-exclusion. Examples 4.5 and 4.10 are good to review here for help as well.

Hope this helps! Let us know if there are additional questions!