Poker with a Pinochle Deck

Date: 6/30/96 at 20:47:58
From: Anonymous
Subject: Pinochle

Imagine that we are playing poker with a pinochle deck of 48 cards: 
two each of 9, 10, J, Q, K, A in each of four suits.  If we a deal a 
5-card poker hand, what is the probability that it is a full house?

Thanking you ahead of time Dr.Math - I appreciate it.

Date: 7/1/96 at 16:38:50
From: Doctor Anthony
Subject: Re: Pinochle

Full house = (for example) 2 aces + 3 jacks

We could pick the face value of the pair of cards (aces in this case) 
in 6 ways, and then the face value of the threesome (jacks in this 
case) in 5 ways.  So face values could be chosen in 6*5 = 30 ways.

Now we must choose 2 aces from 8 and 3 jacks from 8 while there are a 
total of 48 cards from which we must select 5.  So probability of 2 
aces and 3 jacks is  

             (8_C_2)*(8_C_3)/(48_C_5)

and this result must be multiplied by 30 to cover other possibilities 
for the face values of the cards making up the full house.
Required probability = 30*28*56/1712304
                     =  0.02747 

-Doctor Anthony,  The Math Forum
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