Probability of $3+3$ cards, out of $6$ cards drawn from a solitaire 
A solitaire consists of $52$ cards. We take out $6$ out of them (wihout repetition). Find the probability there are $3+3$ cards of the same type (for example, $3$ "1" and $3$ "5").

Attempt. 
First approach. There are $\binom{13}{2}$ ways to choose $2$ out of the $13$ types and by the multiplication law of probability, the desired probability is $$\binom{13}{2}\frac{4}{52}\,\frac{3}{51}\,\frac{2}{50}\,
\frac{4}{49}\,\frac{3}{48}\,\frac{2}{47}.$$
Second approach. There are $\binom{13}{2}$ ways to choose $2$ out of the $13$ types and the desired probability is $$\binom{13}{2}\frac{\binom{4}{3}\binom{4}{3}\binom{4}{0}\ldots\binom{4}{0}}{\binom{52}{6}}.$$
These numbers don't coincide, so I guess (at least) one of them is not correct.
Thanks in advance for the help.

In order to fully clear your confusion, let us tackle a simpler problem first.
We are dealing with drawing w/o replacement, (hypergeometric distribution)
If asked to find the Pr of drawing $2$ red and $3$ blue balls from a pool of $5$ red and $4$ blue balls,
Using the multiplication rule, $P(RRBBB)$ in that particular order$\;= \dfrac59\dfrac48\dfrac47\dfrac36\dfrac25$,
but we would need to multiply it by $\dfrac{5!}{2!3!}$ to take care of all possible orders.
[ But this multiplication factor is all too often forgotten by students]
By the combination approach, we would simply use $\dfrac{\binom52\binom43}{\binom95}$
I would advise that you use direct multiplication of probabilities when a specific order is given, and combinations otherwise.

To come back to your problem, you should be able to see that in your first approach, since there are $3$ each of the two types, you need a multiplier of $\dfrac{6!}{3!3!}$,
thus $\dbinom{13}2\dfrac4{52}\dfrac3{51}\dfrac2{50}\dfrac4{49}\dfrac3{48}\dfrac2{47}\times \dfrac{6!}{3!3!}$
whereas the second approach directly gives the correct answer,
in fact you should simplify it to $\dbinom{13}2 \frac{\binom43\binom43}{\binom{52}6}$ 

For a variety of problems on drawing colored balls from an urn without replacement, you could have a look here
