A five-person committee consisting of students and teachers is being formed to study the issue of student parking privileges. Of those who have expressed an interest in serving on the committee, 12 are teachers and 14 are students. How many ways can the committee be formed if at least one student and one teacher must be included?
What I did: 12C1 * 14C1 * 24C3 = 340032
(Choose out of 12 teachers for one spot, out of 12 students for another, and finally out of 24 people remaining choose 3 for the remaining three spots)
Answer in back of book: 62986
Where am I going wrong?
Best Answer
So you have a total of $26$ people to choose from, teachers and students but only $5$ can be representatives for the team. We need to keep in mind that we can't have all teachers or students in the committee at one time. So what this means is that the number of ways to arrange the teachers is $$C (12,5) = 792$$ The number of ways to choose the students is $$C(14,5) = 2002$$ This is straight from what we are given. Overall, if we assume that we have no restriction at all, we have a total of $$C(26,5) = 65780$$ Since we can't have all teachers or students represent the team, subtract $$65780 - 2002 - 792 = 62986$$ Another reason we subtract from the total is because we don't want to count our arrangements more than once.