Discrete math counting principles

Question

In how many ways can a teacher distribute 12 identical science books among 15 students if

1) no student gets more than one book?

2) if the oldest student gets two books but no other student gets more than one book?

Initially, I tried the binomial theorems with the idea that the student gets a book or no book but that is not giving me the correct solution.

Answer 1

Assuming the students are distinguishable:

For the first one: If no student gets more than one book, then it is just a matter of finding the number of ways to pick 12 students out of those 15, i.e. $15 \choose 12$

For the second one: the oldest one gets 2 books ... leaving 14 students ... and 10 books ... can you do the rest?

Answer 2

Hint for 1) Count the number of subsets of cardinality $12$ of a set of $15$ elements.

Hint for 2) Count the number of subsets of cardinality $12-2$ of a set of $15-1$ elements.