A professor teaching a Discrete Math course gives a multiple choice quiz that has four questions, each with four possible responses: a, b, c, d.

1. A professor teaching a Discrete Math course gives a multiple choice quiz that has four questions, each with four possible responses: a, b, c, d. What is the minimum number of students that must be in the professor's class in order to guarantee that at least two answersheets must be identical? (Assume that no answers are left blank.)

2. You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get a) two pairs (for instance, two aces and two kings), and b) four of a kind (for instance, all four queens).

3. Use the binomial theorem to expand (x + y)6.