Probability of drawing a full house

[Biosyn]

Homework Statement

What is the probability of drawing a full house from a standard deck (52).

Homework Equations

The Attempt at a Solution

These are the two answers I came up with:

(52/52) * (3/51) * (2/50)* (48/49) * (3/48) * 5C3 * 5C2 = 0.0144~

or

(52/52) * (3/51) * (2/50)* (48/49) * (3/48) * 5C3 = 0.00144~


The second answer is the correct one.

I know you multiply by 5 choose 3 because there are three matching cards that you need to select out of five spots.
But what about the pair? Do I need to multiply by 5 choose 2?

The simulation on Wolframalpha comes up with a probability that is close to my second answer.(The correct one).

 

[haruspex]

I know you multiply by 5 choose 3 because there are three matching cards that you need to select out of five spots.
But what about the pair? Do I need to multiply by 5 choose 2?

The multiplication by 5C3 expresses that you don't care which of the 5 cards are the trio. But once that's decided, there are only two spots left for the pair, so you do not want to multiply by 5C2 as well. In fact, you could have done it either way round, 5C3 and 5C2 being the same.

 

[Biosyn]

The multiplication by 5C3 expresses that you don't care which of the 5 cards are the trio. But once that's decided, there are only two spots left for the pair, so you do not want to multiply by 5C2 as well. In fact, you could have done it either way round, 5C3 and 5C2 being the same.

Oh, I see. So after the location of the first three cards are 'picked', the location of the other 2 cards have been determined. Or vice versa where the location of the pairs are determined first and then the three cards basically have no where else to go.
Thanks!