Math 221 Card Problems: Solutions
A standard 52-card deck consists of 13 cards from each of 4 suits (spades, hearts,
diamonds, clubs). The 13 cards have value 2 through 10, jack (J), queen (Q), king (K), or
ace (A). Each value is a “kind” of card. The jack, queen, and king are called “face
cards”.
Note: All of the names of hands listed below are completely fictitious, so your answers
do not need to be mutually disjoint.
1. How many 5-card hands contain a Face Place, i.e. all face cards?
There are 12 total face cards, from which we must choose 5.
12 
   792
5
2. How many 5-card hands contain a Rainbow, i.e. include all four suits?
There must be 2 cards from one suit and 1 card each from the other three suits.
 4  13  3
   13  685, 464
1 2 
3. Suppose the ace of spades is removed from the deck, i.e. now there are only 51 cards
being used. How many 5-card hands (from these 51) are a full house, i.e. three cards
of one kind and two cards of another kind?
There are three disjoint cases: (1) the 3 remaining aces are the three of a kind, (2) 2 of
the aces are the pair, or (3) there are no aces in the hand.
12   4   3  12   4  12   4  11  4 
                     3384
 1  2  2 1  3  1  3 1  2
The following problems involve 6-card hands.
4. How many total 6-card hands are there?
 52 
   20,358,520
6
5. How many 6-card hands contain a Rainbow, i.e. include all four suits?
There are two disjoint cases: (1) 3 cards from one suit and 1 card each from the other
three suits or (2) 2 cards each from two suits and 1 card each from the other two suits.
 4  13  3  4  13  2
   13     13  2,592, 460
1 3 
 2 2 
6. How many 6-card hands contain a Triple Double, i.e. three distinct pairs?
3
13   4 
     61, 776
 3  2
7. Suppose that the queens are “female” cards and that both kings and jacks are “male”
cards. How many 6-card hands contain a Triple Date, i.e. three male cards and three
female cards?
There are 8 total male cards from which we must choose 3, and 4 total female cards
from which we must choose 3.
8 4
     224
 3  3 
8. How many 6-card hands contain a Crowded House, i.e. three cards of one kind, two
cards of a second kind, and one card of a third kind?
13   4  12   4 11  4 
            164, 736
 1   3   1   2  1   1 