Solving the Combinatronics Dilemma

[modulus]

I solved a problem, by two different, and (apparently) viable solutions. But, each solution led me to a different answer:
The question asks me to find out the number of possible ways in which I can choose 5 cards from a standard pack of 52 cards, so that I choose at least a single card from each suite.

FIRST METHOD:
One card from each suite of 13 means a total of [¹³C₁ * ¹³C₁ * ¹³C₁ * ¹³C₁].
After that, I'll have 52-4=48 cards left from which I need to choose only one. That means ⁴⁸C₁.
The final result:
¹³C₁ * ¹³C₁ * ¹³C₁ * ¹³C₁ * 48

SECOND METHOD:
I'll be choosing one card from three suites, but exactly two cards from the fourth suite. So that means:
¹³C₁ * ¹³C₁ * ¹³C₁ * ¹³C₂
= ¹³C₁ * ¹³C₁ * ¹³C₁ * ¹³C₂ * (12/2)
And since I do this with each of the four suites (choose two from it only, while taking one from each of the others), I shall multiply this by 4, giving me:
¹³C₁ * ¹³C₁ * ¹³C₁ * ¹³C₂ * 24


And this is half of the answer arrived at by the first method...
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what went wrong?

 

[Office_Shredder]

In the first method you double count. For example I could have picked my cards like this:

ace of hearts, ace of spades, ace of clubs, ace of diamonds

and then the king of heartsOR

king of hearts, ace of spades, ace of clubs, ace of diamonds

and then the ace of hearts

In the first method you count those as two distinct selections even though they're the same set of five cards

 

[modulus]

Oh...I see.
Thank you.