(a) A bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. What is the probability that, when he stops, there is at least 1 red ball and at least 1 green ball on the table?

(b) Suppose that f(a) = 2a² 3a + 1 for all real numbers a and g(b) = log 1/2 b for all b > 0. Determine all with 0 2 for which f(g(sin )) = 0.

(c) (b) Prove that the integer ((1!)(2!)(3!) (398!)(399!)(400!)) / 200! is a perfect square. (In this fraction, the numerator is the product of the factorials of the integers from 1 to 400, inclusive.)