**Poker Digest Vol. 2, No. 3, January 29 - February 11, 1999**

In my previous article I gave a table for the number of high hands in mambo stud and promised to discuss the computation of the numbers. Let me remind you we are counting the number of three-card hands from all possible 270,725 four-card hands; the mambo hands are straight flush, three-of-a-kind, straight, flush, one pair, high card; and the four-card hands are four-of-a-kind, four-card straight flush, four-card flush containing a straight flush, four-card flush not containing a straight flush, four-card straight containing a straight flush, four-card straight not containing a straight flush, straight flush with a pair, straight flush without a pair, three-of-a-kind, straight and flush, straight without a pair, straight with a pair, flush without a pair, flush with a pair, two pair, one pair and high card.

Some readers may wonder why I advocate dividing up the four-card hands as I have done rather than simply counting the three-card hands directly. Let me illustrate why with three-of-a-kind hands. The four-card hands which count as three-of-a-kind when evaluated as three-card hands are four-of-a-kind and three-of-a-kind. So the number is the sum of the number of four-of-a-kind hands and the number of three-of-a-kind hands.

There are 13 choices for the rank of a four-of-a-kind and the four cards are determined at that point so there are 13 four-of-a-kind hands. To count three-of-a-kind hands, there are 13 choices for the rank, four choices for three cards of that rank, and 48 choices for the remaining card. This gives us three-of-a-kind hands. Adding 13 to the latter number gives us 2,509 hands which are three-of-a-kind hands and this is the number in the table of the previous article.

Now suppose we try to ``directly'' calculate the number of four-card hands which will be considered three-of-a-kind hands. There are 13 choices for the rank and four choices for the three-of-a-kind of that rank. This leaves 49 choices for the remaining card. Multiplying gives . Of course, this is incorrect as we have seen. Why? I can only ask the reader to think about it and try to see that four-of-a-kind hands are counted four times in the last calculation. So we must subtract 39 to compensate for the overcounting and this produces 2,509 which is correct. Now if you say to me the problem occurs because of multiplying by 49 and all we have to do is consider the possibilities when choosing the last card. But as soon as you start making provisions for the choice of the last card, you are essentially dividing the computation into two parts which is equivalent to partitioning the four-card hands into four-of-a-kind hands and three-of-a-kind hands. Why not just do it upfront and reduce the chance of errors?

I shall now run through the remaining four-card hands and provide details for a few of the more interesting cases.

**Four-card straight flush.**- There are 44 four-card straight flushes.
**Four-card flush containing a straight flush.**- There are eight straight flushes
beginning with an A or a Q. Any of nine cards can be added to each to produce
a flush which is not a four-card straight flush. There are 40 straight
flushes remaining and any of eight cards can be added to each. This gives us
four-card flushes containing straight flushes.
**Four-card flush not containing a straight flush.**- We employ a technique
in this case which will be used in several other cases leading us to explain
it now in detail. There are
*C*(13,4) = 715 sets of four distinct ranks one can choose from 13 ranks. Of these, 11 have the form . Another 98 have the form , where*y*is neither*x*-1 nor*x*+3, for if*x*= A or*x*= Q, there are nine choices for*y*and if*x*lies between deuce and J, inclusive, there are eight choices for*y*. So removing these 109 sets of ranks leaves 606 sets of ranks which contain no straight or four-card straight possibilities. Thus, for each of these 606 sets of ranks, there are four choices of suits giving us 2,424 four-card flushes containing neither a straight flush nor a four-card straight flush. **Four-card straight containing a straight flush.**- There are 264 four-card straights containing a straight flush.
**Four-card straight not containing a straight flush.**- There are 11 sets
of ranks corresponding to four-card straights. There
are 4
^{4}= 256 choices of the four cards, but some choices correspond to hands already counted: All in the same suit is a four-card straight flush, and either the first three or the last three in the same suit gives a straight flush. There are four choices for the former and 24 choices for the latter. Removing these 28 hands already counted gives such hands. **Straight flush with a pair.**- There are 432 straight flushes which also contain a pair.
**Straight flush without a pair.**- If a straight flush begins with
an ace or queen, any of 27 cards may be added without forming any hand
which already has been counted. If a straight flush begins with any other
card, one may add any of 24 cards without creating a hand already counted.
This produces
straight flushes with no pairs.
**Straight and flush.**- This is a hand of the form
*x*,*x*+1,*x*+2,*y*, where precisely two of the cards from ranks*x*,*x*+1,*x*+2 are in the same suit as*y*and . We saw above that there are 98 such sets of four ranks. There are four choices for the suit of*y*, three choices for the other two cards of the same suit, and three choices for the other suit. This gives us such hands. **Straight without a pair.**- There are 19,992 straights without a pair.
**Straight with a pair.**- The set of ranks for this type of hand is
.
There are 12 such sets. There are three choices for which
rank is paired and six choices for the pair of the chosen rank. The suits
for the remaining two ranks cannot both agree with one of the suits of the
pair or we would have a straight flush. Hence, there are 16 - 2 = 14
choices for the other two cards. This gives us
straights with a pair.
**Flush without a pair.**- As we saw in an earlier case, there are 606
sets
of ranks not admitting four-card straights or straights. Since four-card
flushes have been counted already, we choose three of the ranks
to be suited in four ways, there are four choices for the suit, and
three choices for the remaining suit. We obtain
flushes without a pair.
**Flush with a pair.**- There are 9,864 flushes with a pair.
**Two pair.**- There are 2,808 hands of two pair.
**One pair.**- There are 274 sets
of ranks not allowing a straight. There are three choices for the
rank of the pair, six choices
for the pair and 14 choices for the remaining 2 cards since we cannot allow
both of the remaining 2 cards to be in the same suit as either suit of the
pair as this would produce a flush. Therefore, there are
hands with a single pair.
**High card.**- There are 606 sets of ranks not allowing a straight or four-card straight. We also must eliminate flushes. Of the 256 choices for the cards, four are all in the same suit and 48 have three in the same suit. Eliminating them leaves 204 choices for each set of ranks. This produces hands of this type.

From the above information, one easily can obtain the entries of the table in my previous article by coalescing the appropriate four-card hands corresponding to the same type of three-card hand.

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