What is the probability of drawing a full house on the first five cards drawn in a 52 card deck?

A full house occurs when you pick a pair and a three of a kind.

Lets say the pair cards are P and the three of a kind cards are T.

You have PPTTT, in all the possible arrangements. Can you find the number of arrangements?

Then, the probability of getting a card in the P slot is 52 (52 possible cards)
The probability that you pick a card to put in the second P slot, so that this card makes a pair with the first car is 3C1 (only 1 card of the 3 remaining same sign card)

You proceed the same way with the three of a kind. There are 48 possible cards left though.

Then find all the ways of getting 5 cards from 52 cards.

Divide the required events by the number of ways to get 5 cards from 52 cards.