**Brian Alspach**

**25 August 2000**

We examine properties of boards with respect to ranks and suits.

It is well known that a set contains distinct elements, but there are
times when it is convenient to have an element repeated. In this case
we use the term *multiset*. The *multiplicity* of an element
is the number of times it appears in the multiset.

The concept of multiset comes into play when discussing boards in
either hold'em or Omaha. The *rank multiset* of a board is the
multiset of all ranks in the board, whereas, the *rank set* of a
board is the set of all ranks in the board. An example makes this
clear. Suppose the board consists of the cards
.
Then the rank multiset of the board
is
and the rank set is .

We first examine boards with respect to ranks. There are six types of rank multisets for boards.

**Type A.**- Type A is a rank multiset of the form
,
that is, the ranks are distinct. The number of Type A rank multisets
is given by
.
To obtain the number of Type A
boards, multiply 1,287 by 4
^{5}because there are 4 choices for each of the 5 ranks. Doing so produces 1,317,888 Type A boards. **Type B.**- Type B is a rank multiset of the form , that is, the board has a single pair. The number of Type B rank multisets is given by because there are 13 choices for the rank of the pair and the remaining 3 ranks are being chosen from 12 ranks. To obtain the number of Type B boards, multiply the preceding number by because there are 6 choices for the pair of the given rank, and 4 choices for each card of the remaining 3 ranks. This leads to 1,098,240 Type B boards.
**Type C.**- Type C is a rank multiset of the form , that is, the board has two-pair. The number of such rank multisets is given by because there are 13 choices for the rank of the singleton and we are choosing the 2 ranks of the pairs from 12 ranks. The number of Type C boards is obtained by multiplying the preceding number by because there are 6 choices for each of the pairs of the given ranks, and 4 choices for the card of the singleton rank. This yields 123,552 boards.
**Type D.**- Type D is a rank multiset of the form , that is, the board is a 3-of-a-kind hand. The number of Type D rank multisets is because of 13 choices for the rank of the 3-of-a-kind, and 2 choices from 12 for the ranks of the 2 singletons. To obtain the number of Type D boards, we multiply the preceding number by because there are 4 choices for the 3-of-a-kind of the given rank, and 4 choices for each of the cards of the other 2 ranks. This produces 54,912 Type D boards.
**Type E.**- Type E rank multisets have the form , that is, the board contains quads. There are 13 choices for the rank of the quads and 12 choices for the rank of the singleton. This yields 156 Type E rank multisets. There are 4 choices for the card of the singleton rank leading to Type E boards.
**Type F.**- Type F rank multisets have the form
,
that is, there is a full house on board. There are 156 Type F rank
multisets with 13 choices for
*x*and 12 choices for*y*. There are 4 choices for the trips of rank*x*and 6 choices for the pair of rank*y*. Hence, there are 3,744 boards of Type F.

The preceding information is contained in the following table. The column head ``prob'' is the probability of a board occurring of that particular type.

RANK DISTRIBUTION

Type | Rank Multisets | Boards | Prob |

A | 1,287 | 1,317,888 | .507 |

B | 2,860 | 1,098,240 | .423 |

C | 858 | 123,552 | .048 |

D | 858 | 54,912 | .021 |

E | 156 | 624 | .0002 |

F | 156 | 3,744 | .0014 |

Now we move to considering suit distributions. We shall label these boards using a vector notation.

**(2,1,1,1).**- A Type (2,1,1,1) suit multiset is a multiset in which 2 cards are in the same suit and the remaining 3 cards are distributed with 1 from each of the remaining 3 suits. There are 4 choices for the suit which appears twice so that there are 4 multisets of this type. There are choices for the cards of the suit with 2 cards and 13 choices for each of the remaining cards. Thus, there are such choices. Multiplying by 4 yields 685,464 boards of Type (2,1,1,1).
**(2,2,1,0).**- A Type (2,2,1,0) suit multiset has 2 suits represented with 2 cards apiece and 1 suit with a singleton. There are 6 choices for the 2 doubleton suits and 2 choices for the singleton suit. Thus, there are 12 multisets of Type (2,2,1,0). There are 78 choices for the 2 cards of a doubleton suit, and 13 choices for the card of the singleton suit. This gives 79,092 choices. Multiplying by 12 yields 949,104 boards of Type (2,2,1,0).
**(3,1,1,0).**- A Type (3,1,1,0) suit multiset has 3 cards of 1 suit and a card each of 2 other suits. There are 4 choices for the suit with 3 cards and 3 choices for the remaining 2 suits chosen from 3. This gives us 12 multisets of Type (3,1,1,0). There are choices for the 3 cards from the given suit and 13 choices for each of the remaining 2 cards. That produces choices. Multiplying by 12 gives us 580,008 Type (3,1,1,0) boards.
**(3,2,0,0).**- A Type (3,2,0,0) suit multiset has 3 cards from 1 suit and 2 cards from another suit. There are 12 such suit multisets. There are 286 choices for the 3 cards of the one suit, and 78 choices for the 2 cards of the other suit. This gives us Type (3,2,0,0) boards.
**(4,1,0,0).**- A Type (4,1,0,0) suit multiset has 4 cards from 1 suit and another card from a different suit. There 12 multisets of this form. There are choices for the 4 cards from the same suit and 13 choices for the remaining card. Altogether this produces boards of Type (4,1,0,0).
**(5,0,0,0).**- Finally, a Type (5,0,0,0) suit multiset has all 5 cards from the same suit. There are 4 such multisets and choices for the 5 cards from the suit. This yields 5,148 Type (5,0,0,0) boards.

As before we place the preceding information in a table. The column headed `prob' is the probability of a board of that type occurring.

SUIT DISTRIBUTION

Type | Suit Multisets | Boards | Prob |

(2,1,1,1) | 4 | 685,464 | .264 |

(2,2,1,0) | 12 | 949,104 | .365 |

(3,1,1,0) | 12 | 580,008 | .223 |

(3,2,0,0) | 12 | 267,696 | .103 |

(4,1,0,0) | 12 | 111,540 | .043 |

(5,0,0,0) | 4 | 5,148 | .002 |

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