Imagine you draw the five cards, but instead of looking at any of them you just put them face-down on the table. Now, turn over card number two and three and look at their suit. This is the same as taking the top card, putting it on the bottom (which could be seen as part of the shuffle process), and leaving the fourth and fifth card on top of the deck.
What I'm trying to get to here is that you can ignore the three other cards. It's not easy to accept, and I therefore show you the calculations on an easier example, like this: If I draw two cards, what is the probability that the second card is a diamonds?
If you do it the hard way, you have to take into account the first card, and you get $$ \frac{13}{52}\cdot \frac{12}{51} + \frac{39}{52} \cdot \frac{13}{51} $$ but this turns out to be $\frac{1}{4}$, just as if you'd _completely ignored_ the first card. You can do the same in your example.