Generating function for number of ways n people can pick a total of r1 chairs of type 1, r2 chairs of type 2 etc This is a homework question for my combinatorics class that I just need to be pointed in the right direction to start. 
Find a generating function $x_1, x_2, . . . , x_m$ whose coefficient of $x_1^{r_1} x_2^{r_2} . . . x_m^{r_m}$ is
the number of ways $n$ people can pick a total of $r_1$ chairs of type $1$, $r_2$ chairs of type $2$, . . . $r_m$ chairs of type $m$ if (a, b, and c are all unique and separate scenarios)
(a) Each person picks one chair
(b) Each person picks either two chairs of one type or no chairs at all
(c) Person $i$ picks up to $i$ chairs of exactly one type
 A: Basic idea: [I am solving only part a]
a)

Let $P(r_1,r_2,...,r_m)$ denote the number of ways n people can pick a total of $r_1$ chairs of type 1, $r_2$ chairs of type 2, . . . $r_m$ chairs of type m such that each person picks one chair. Clearly, $$n=r_1+r_2+...+r_m$$
Claim
$$P(r_1,r_2,...,r_m) = {n\choose r_1}*{(n-r_1)\choose r_2}...{(n-(r_1+r_2+...+r_{m-2})\choose r_{m-1}}*{r_m\choose r_m}  $$
$$= n! / (r_1!r_2!...r_m!) = {n \choose r_1,r_2,..,r_m}$$[Known as multinomial coefficient]
Proof :You select $r_1$ people from n who will receive chair of type 1 in ${n\choose r_1}$ ways, next you select $r_2$ people from remaining ones who will receive chair of type 2 in ${(n-r_1)\choose r_2}$ ways and so on.

Thus your generating function F is, $$ F(x_1,x_2,..,x_m) = \sum_{r_1+r_2+...+r_m=n}P(r_1,r_2,...,r_m)*x_1^{r_1}*x_2^{r_2}...*x_2^{r_m}$$
 $$=\sum_{r_1+r_2+...+r_m=n}{n \choose r_1,r_2,..,r_m}*x_1^{r_1}*x_2^{r_2}...*x_m^{r_m} = (x_1+x_2+...+x_m)^n$$
This is also known as multinomial distribution. You can find out the generating function for b) and c) in similar way.
