TITLE: You are dealt five cards from a standard deck. What is the probability that you'll have at most three kings in your hand?
QUESTION [6 upvotes]: You are dealt five cards from a standard and shuffled deck of playing cards. Note that a standard deck has 52 cards and four of those are kings. What is the probability that you'll have at most three kings in your hand? 

I know that the answer is $\frac{54144}{54145}$ from the answer key, and I know that the sample space is ${^{52}\mathrm C_5}$. What I don't get is how to find the event. 
Do I just add the combination for each number of kings together (${^5\mathrm C_2} + {^5\mathrm C_3}$)?
Or do I need to multiply as well to account for the other cards in the 5-card hand?

REPLY [3 votes]: Yet another way: there are 52 cards in the deck, 4 of which are kings.  The probability the first card is a king is 4/52= 1/13.  There are then 51 cards left, 3 of them kings.  The probability the second card is a king is 3/51= 1/17.  There are then 50 cards left, 2 of them kings.  The probability the third card is a king is 2/50= 1/25.  There are then 49 cards left, 1 of them a king.  The probability the fourth card is a king is 1/49.  There are no longer any kings in the deck so the probability the fifth card is not a king is 1.
The probability of getting "four kings and one non-king" in that order is (1/13)(1/17)(1/25)(1/49).  There are 5 ways to order "four kings and one non- king" (the non-king being in any of the 5 places) so the probability of "four kings and one non-king" in any order is 5(1/13)(1/17)(1/25)(1/49).
Finally, the probability of get "at most three kings in a five card hand" is 1- 5(1/13)(1/17)(1/25)(1/49).