A student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
Let's first look at case 1. Our first step is to choose the math course. That's right, we have 2 choices: algebra or geometry. So we have $2\binom{3}{2}=6$ possible programs for case 1. Now let's move to case 2. We have to choose 2 Math courses. That's right, we have $\binom{2}{2}$ choices for Math. So we have $\binom{2}{2}\binom{3}{1}=3$ possible programs for case 2. Lastly we have to add the number of programs from each case. So we have $6+3=9$ possible programs. That's right. We have 9 possible programs.

