Exercise 19

A teacher has $450$ identical pieces of candy. He wants to distribute them to his class of $65$ students, although he is willing to take some leftover candy home. (He does not insist on taking any candy home, however.) The student who won a contest in the last class is to receive at least $10$ pieces of candy as a reward. Of the remaining students, $34$ of them insist on receiving at least one piece of candy, while the remaining $30$ students are willing to receive no candy.

1. In how many ways can he distribute the candy?

2. In how many ways can he distribute the candy if, in addition to the conditions above, one of his students is diabetic and can receive at most $7$ pieces of candy? (This student is one of the $34$ who insist on receiving at least one piece of candy.)