For 1: You have 5 math teachers, and you want to choose two of them. Using binomial coefficient notation, there are ways to choose the two math teachers. Next, you want three English teachers. The choice of English teacher is independent from the choice of math teacher. So, you can apply the product principle. There are ways to choose the three English teachers. The total number of committees is given by the product of those two.
2. For the first ball, your chances of drawing a white ball are . Now there is one less ball in the mix. Your chances of drawing a black ball are . Finally, the probability that the final ball is green is . So, by the product principle, the probability is the product of those three probabilities.