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.

413 3 13 685,464 12

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 12  213 1312 

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.

41332  413 13  13 2,592,460 13  22

6. How many 6-card hands contain a Triple Double, i.e. three distinct pairs?

3 13 4  61,776 32

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.

84  224 33

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 131211 