Probability of cards drawn from a deck

Question

The cards are drawn from a well shuffled deck of 52 cards one after the other without replacement. The probability of first card being a spade and the second a black king is ?

Here, is my approach, Upon first draw we got a black spade king $$P(\text{first card is spade}) = \frac{13}{52}$$ $$ P(\text{second black king}) = \frac{1}{51} $$

Upon first draw we don't get a black spade king $$P(\text{first card is spade}) = \frac{13}{52}$$ $$ P(\text{second black king}) = \frac{2}{51} $$ Now, the total probability, $$P= \frac{13}{52} \frac{1}{51} + \frac{13}{52} \frac{2}{51} = \frac{39}{2652}$$

But the actual answer is $\frac{25}{2652}$

Answer 1

Upon first draw we got a black spade king $$P(\text{first card is $K \spadesuit$}) = \frac{1}{52}$$ $$ P(\text{second black king i.e $K\clubsuit$}) = \frac{1}{51} $$

Upon first draw we don't get a black spade king $$P(\text{first card is $\spadesuit$ but not $K\spadesuit$}) = \frac{12}{52}$$ $$ P(\text{second black $K$}) = \frac{2}{51} $$ Now, the total probability, $$P= \frac{1}{52} \frac{1}{51} + \frac{12}{52} \frac{2}{51} = \frac{25}{2652}$$

Answer 2

As one card is common in both.

So probability = $\frac{13}{52} \cdot \frac{2}{51} -\frac{1}{52 \cdot 51}$

= $\frac{26}{2652} -\frac{1}{2652}$

= $\frac{25}{2652}$