How many different ways can a group of students be hired to work a survey?

Question

Dear Math Stack Exchange,

I understand we are not supposed to ask for answers and so I am not asking you to solve this question. I was just wondering if anyone knew how to go about these types of problems because I am so lost I cannot even explain the question without copy and pasting it.

The Statistics Survey Centre wishes to hire some undergraduates to help in a phone survey. On the call list are 6 second-year students, 3 third-year and 3 fourth-year students. Policy is to always hire at least one third year and one fourth year student. Other than that as many students as are needed can be hired without restriction.

(Note: You are being asked how many different groups could be hired, that is how many ways is it possible to select a group of students given the conditions?)

Answer 1

Generating functions are useful in this type of problem \begin{eqnarray*} (1+6x+15x^2+20x^3+15x^4+6x^5+x^6)(3y+3y^2+y^3)(3z+3z^2+z^3) \end{eqnarray*} The coefficient of $x^a y^b z^c$ will give the number of ways to have $a$ second year students, $b$ third years and $c$ fourth years.

Answer 2

There are $2^{12}$ ways to choose students.

There are $2^9$ ways to choose students without choosing anyone that's third year.

There are $2^9$ ways to choose students without choosing anyone that's forth year.

There are $2^6$ ways to choose students without choosing anyone that's forth year or third year.

Therefore there are $2^{12}-2\times2^9+2^6=3136$ ways.