Sampling four socks from $n$ distinct pairs of socks 
A drawer contains $n$ distinct pairs of socks. A sample of four socks is made, without replacement. Let $X$ denote the number of pairs of socks in the sample. What is the probability mass function of $X$?

The variable $X$ can take on only the values $0$, $1$, and $2$, so I compute the probability that $X$ takes on each value.
For $P(X = 0)$, all the socks have to come from different pairs, so it suffices to multiply the four probabilities, that each new chosen sock does not form a pair with any previously chosen sock.
$$
P(X=0)
= 1 
\cdot \frac{2n - 2}{2n - 1} 
\cdot \frac{2n - 4}{2n - 2} 
\cdot \frac{2n - 6}{2n - 3}
$$
For $P(X=1)$, I focus on a single pair. Say they have a particular color $c$. What is the probability that the sample contains the pair $c$? Let $Y_c$ be the number of  socks of color $c$ in the sample. Of course, $Y_c \leq 2$. The variable follows a hypergeometric distribution. Hence, the probability of getting a pair of color $c$ is
$$
P(Y_c = 2)
= \frac{\binom{2}{2} \binom{2n - 2}{4 - 2}}{\binom{2n}{4}}
= \frac{6}{n(2n-1)}
$$
The probability that there is at least one pair in the sample is obtained by multiplying by $n$.
How can I find the probability that there is exactly one pair and the probability that there are exactly two pairs?

I would approach this combinatorially.
The total number of choices is $\binom{2n}{4}=\frac{2n(2n-1)(2n-2)(2n-3)}{24}=\frac{n(n-1)(2n-1)(2n-3)}{6}$.
For there to be exactly one pair, you choose one of the $n$ pairs in $\binom{n}{1}=n$ ways, then you must choose two of the remaining pairs in $\binom{n-1}{2}=\frac{(n-1)(n-2)}2$ and finally, for each of those two pairs, you must choose one of the two socks in $\binom{2}{1}=2$ ways. Hence, we have $n\cdot\frac{(n-1)(n-2)}2\cdot2^2=$ $2n(n-1)(n-2)$ possibilities, and the probability is
$$\mathbb{P}(X=1)=\frac{2n(n-1)(n-2)}{\frac{n(n-1)(2n-1)(2n-3)}{6}}=\frac{12(n-2)}{(2n-1)(2n-3)}$$
For there to be exactly two pairs, you must choose two of the $n$ pairs in $\binom{n}2=\frac{n(n-1)}2$ and take both socks in the pair, which entirely defines the choice. Hence, the probability is
$$\mathbb{P}(X=2)=\frac{\frac{n(n-1)}{2}}{\frac{n(n-1)(2n-1)(2n-3)}{6}}=\frac{3}{(2n-1)(2n-3)}$$
