Two cards are drawn together from a pack of $52$ cards. What is the probability that one is spade and other is king? I did it from conditional probability, first the case when we pick one spade first and then pick a king next. So it is equal to the probability of picking up a spade*probability of picking a king given that we have already picked up a spade so it is equal to $\frac{13}{52} * \frac{4}{51}$ And then the probability of picking up a king and a spade $=$ probability of picking up a king*probability of picking up a spade given that we have already picked up a king which is equal to $\frac{4}{52} * \frac{13}{51}$ Now summing up both of these cases, what is wrong in this method ?

Spadesuit has its own king. So in the $2$nd case if you pick the king of spades, the number of remaining spades is $12$ not $13$. Your $1$st case has a similar error.

You can choose a non-spade king in $3$ ways and for each choice a spade can be chosen in $13$ ways. If you pick the king of spades, another spade can be chosen in $12$ ways. So the required probability is $$ {3\times 13+12\over\binom{52}{2}} $$