Hi, Gregg:
You seem to have a good grasp of the basic methods for counting
permutations and combinations, so I've marshalled the arguments
below in a somewhat brisk fashion. Please let me know if some or
all of the arguments require more discussion.
I'll present the results as frequency counts within a universe of
53^5 possible outcomes. See the table at bottom for a summary of
the various results.
I've assumed the Ace can be used either in a high or low position
within a straight, as is the case here:
https://winnersplay.com/casino/jokerswild.html
regards, mathtalk
<<<<<<<<<<<<<<<<<< BEGIN Analysis <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
1. Five Jokers : 1 (obvious)
2. Five of a kind suited : 1612
There are 31 proper subsets of the five reels (not all five reels)
which could be occupied by Jokers, and 52 choices for an identical
non-Joker card to occupy any remaining reel. 52 * 31 = 1612
As Gregg has observed, all four Joker hands are classified here.
The total can be expanded into cases by number of Jokers, so:
52 * (C(5,0) + C(5,1) + C(5,2) + C(5,3) + C(5,4))
= 52 + 260 + 520 + 520 + 260
where C(5,k) is combinations of 5 things taken k at a time, shows us
for the case of k Jokers how many five a kind suited outcomes there
are.
3. Straight flush : 43520
Let's break the analysis into cases by number of Jokers. Note that
an Ace in my interpretation may be either the high or low card, but
otherwise no "wrapping around" is allowed in forming a straight.
In each case there's a factor of 4 for choice of suit as all non-
Joker cards must be of the same suit here.
a) no Jokers
The low card of the straight is one of ten possible values (Ace to
10). The five cards are distinct so there are 5! arrangements of
these on the reels:
10 * 4 * 5! = 4800
b) one Joker
The four non-Joker cards must either be a run of four in sequence,
or else have a missing "inside" card filled by the Joker. We can
diagram and enumerate the possibilities by choices for low card:
c1 c2 c3 c4 [11 choices for low card, Ace through Jack]
c1 __ c3 c4 c5 [10 choices for low card, Ace through 10]
c1 c2 __ c4 c5 [10 choices for low card, Ace through 10]
c1 c2 c3 __ c5 [10 choices for low card, Ace through 10]
So there are 41 possibilities for the four non-Jokers. With five
distinct cards there are 5! arrangements of these on the reels:
41 * 4 * 5! = 19680
c) two Jokers
Again we can diagram and enumerate the possibilities according to if
and where the Jokers are used to "fill" an inside straight:
c1 c2 c3 [12 possible low cards, Ace through Queen]
c1 __ c3 c4 [11 possible low cards, Ace through Jack]
c1 c2 __ c4 [11 possible low cards, Ace through Jack]
c1 __ __ c4 c5 [10 possible low cards, Ace through 10]
c1 __ c3 __ c5 [10 possible low cards, Ace through 10]
c1 c2 __ __ c5 [10 possible low cards, Ace through 10]
So there are 64 possibilities for the three non-Jokers. Since the
two Jokers are interchangeable, there are 5!/2 = 60 arrangements on
reels:
64 * 4 * (5!/2) = 15360
d) three Jokers
With these diagrams:
c1 c2 [13 possible low cards, Ace through King]
c1 __ c3 [12 possible low cards, Ace through Queen]
c1 __ __ c4 [11 possible low cards, Ace through Jack]
c1 __ __ __ c5 [10 possible low cards, Ace through 10]
So there are 46 possibilities for the two non-Jokers. The three
Jokers are interchangeable, so there are 5!/3! = 20 arrangements on
the reels:
46 * 4 * (5!/3!) = 3680
Combine these case by case numbers to get the total straight flushes:
4800 + 19680 + 15360 + 3680 = 43520
Also, keep the diagrams above in mind when we count the "simple"
straight outcomes.
4. Five of a kind unsuited : 39000
We will handle this by showing that the total number of five of a
kind hands, whether suited or not, is 13 * (5^5 - 1). Then we just
subtract the previously found number 1612 of five of a kind suited
hands to get the number 39000 of five of kind unsuited hands.
Since we exclude the five Joker hand from this count, any five of a
kind hand, whether suited or not, has a well defined denomination.
There are 13 possible choices for the denomination. Given that
choice of denomination, any particular reel not occupied by a Joker
must have one of four possible cards.
If you consider the number k of Jokers ranging from 0 to 4, then an
expression which counts the number of five of a kind outcomes is:
13 * [4^5 + 4^4*5 + 4^3*C(5,2) + 4^2*C(5,3) + 4*C(5,4)]
Here the first term represents using no Jokers, the second using one
Joker, and so forth. The combinations C(5,k) for k Jokers is of
course consistent with choosing k of 5 reels to be Jokers; thus by
comparison with binomial expansion of (4+1)^5, the count is:
13 * (5^5 - 1) = 40612
After subtracting five of kind suited's, 40612 - 1612 = 39000.
5. Four of a kind suited : 183520
The case analysis by number of Jokers is a little more complicated,
but we are able to close out the rest of the three Joker hands.
a) no Jokers
Pick one card (suit and denomination) to be the four of kind, and
any different denomination card to be the fifth (since otherwise the
fifth card having equal denomination would give five of a kind).
Four cards are interchangeable, so for the outcomes we have five
possible reel arrangements:
52 * 48 * 5 = 12480
For future reference we'll note that one-quarter of these turn out
to be flushes if the suit of the fifth card agrees with the others.
b) one Joker
Even with a wildcard the suit and denomination of the four of a kind
are identifiable. Choose that, and the fifth card of a different
denomination, and we have three interchangeable cards.
Thus the number of reel arrangements is 5!/3! = 20:
52 * 48 * 20 = 49920
Again we'll note for future reference that one-quarter of these are
also flushes.
c) two Jokers
We choose the suit and denomination for an identical pair and some
fifth card with different suit or denomination. The Joker pair and
non-Joker pair are each interchangeable, so we have 5!/(2^2) = 30 as
the number of reel arrangements:
52 * 48 * 30 = 74880
Again we'll note for future reference that one-quarter of these are
also flushes.
d) three Jokers
With three Jokers and any two non-Joker cards, one gets four of a
kind suited or better. In fact we can compute the outcomes in this
case simply by subtracting from the number of all three Joker "hands"
(outcomes) those which have already been counted above as straight
flushes or five of a kinds (suited or unsuited).
There are C(5,3) = 10 choices for reels to hold three Jokers, and
52^2 possibilities for the other two reels in an outcome with exactly
three Jokers:
10 * 52^2 = 27040
Previously we counted 4*920 = 3680 of these as straight flushes, and
within five of a kind categories we counted 13 * 4^2 * C(5,3) three
Joker hands (see the expression under five of a kind unsuited for
details). Thus the residue of three Joker hands which will only
qualify as four of a kind suited is:
27040 - (3680 + 2080) = 21280
Collecting the results from our case analysis:
12480 + 49920 + 74880 + 21280 = 158560
gives the total number of four of kind suited outcomes.
6. Four of a kind unsuited : 1759680
As we did with the five of a kind unsuited category, we come count
the four of a kind unsuited outcomes by subtracting the suited four
of a kind outcomes from the number of all four of a kind outcomes.
It is perhaps a little easier to follow when details are broken down
into cases by numbers of Jokers:
a) no Jokers
Choose a denomination 13 ways, and then a different denomination for
the fifth card in 12 ways. The reel/position for this fifth card is
chosen in 5 ways. Arbitrarily choosing suits for all five reels:
13 * 12 * 5 * 4^5 = 798720
We subtract from this the number 12480 of four of a kind suited with
no Joker outcomes, and we have:
798720 - 12480 = 786240
four of a kind unsuited with no Joker outcomes.
b) one Joker
Choose a denomination 13 ways, and then a different denomination for
the fifth card in 12 ways. The reel/position for this fifth card is
chosen in 5 ways and for the Joker in 4 remaining ways. Arbitrarily
choosing suits for all four non-Joker reels:
13 * 12 * 20 * 4^4 = 798720
which, by a quirk of arithmetic, is the same number as we got above
with no Jokers. We subtract from this the number 49920 of four of a
kind suited with one Joker outcomes, and we have:
798720 - 49920 = 748800
four of a kind unsuited with one Joker outcomes.
c) two Jokers
Choose a denomination (for the pair) 13 ways, the other denomination
in 12 ways, the positions of the Jokers in C(5,2) = 10 ways and of
the "fifth" card in 3 ways. The suits are assigned arbitrarily to
all three non-Joker reels:
13 * 12 * 30 * 4^3 = 299520
Subtract from this the number 74880 of four of a kind suited with two
Jokers outcomes, and you have:
299520 - 74880 = 224640
four of a kind unsuited with two Jokers outcomes.
Collecting the results of these three cases gives:
786240 + 748800 + 224640 = 1759680
Note that because these are the unsuited outcomes, none are flushes.
7. Full House : 2196480
a) no Jokers
Pick the denomination for three of kind (13 ways) and another for a
pair of different denomination (12 ways). Pick the two reels which
the pair will occupy in C(5,2) = 10 ways. Assign suits arbitrarily:
13 * 12 * 10 * 4^5 = 1597440
Note for future reference that 1/256 of these are flushes.
b) one Joker
The non-Joker cards must consist of two pairs, since otherwise we'd
use the Joker with the natural three of a kind to get four of a kind.
Pick the denominations of the two pairs in C(13,2) ways. Pick the
reel for the Joker in 5 ways and the reels for the higher of the two
pairs (counting Ace high for this purpose) in C(4,2) = 6 ways. Suits
are assigned arbitrarily to the non-Jokers:
78 * 5 * 6 * 4^4 = 599040
Note for future reference that 1/64 of these are flushes.
c) two Jokers
This can't happen. With two Jokers we'd still need a least a pair
among the non-Jokers to construct a full house, since you cannot have
three different denominations in a full house hand. But then we'd
put the two Jokers with the pair to get four of a kind instead.
Combining results gives 1597440 + 599040 = 2196480 full houses.
8. Flush : 2053680
Briefly we reduce the total number of flushes by counts previously
covered in preceding categories.
a) no Jokers
Pick a suit (4 ways) and then any of the 13 cards in that suit for
each reel, ie. 4 * 13^5. However we must reduce this total by any
flushes with no Jokers already accounted for above. This includes
all of the five of a kind suited and straight flushes with no Joker,
one quarter of the four of a kind suited with no Joker, and 1/256
of the full houses with no Joker.
4*13^5 - 52 - 4800 - 12480/4 - 1597440/256 = 1470960
b) one Joker
Pick a suit (4 ways), then any of the 13 cards in that suit for each
of four reels, and multiply by 5 for the choice of reel for a Joker
to occupy, ie. 4 * 13^4 * 5. This total must be reduced by those
flushes with one Joker previously accounted for as five of a kind
suited, straight flushes, four of a kind suited, or full houses.
4*13^4*5 - 260 - 19680 - 49920/4 - 599040/64 = 529440
c) two Jokers
Pick a suit (4 ways), then any of the 13 cards in that suit for each
of three reels, and multiply by C(5,2) = 10 for choices of two reels
for the two Jokers to occupy, ie. 4 * 13^3 * 10. This total is then
reduced by flushes with two Jokers previously counted as five of a
kind suited, straight flushes, or four of a kind suited.
4*13^3*10 - 520 - 15360 - 74880 = 53280
Combining these cases gives the total outcomes counted as flushes:
1470960 + 529440 + 53280 = 2053680
9. Straight : 2654400
We reduce the total number of straights by previously recorded counts
of straight flushes.
a) no Jokers
As with straight flushes, the lowest card's denomination is one of 10
values. Assignment of suits to cards is arbitrary, and the distinct
denominations give 5! arrangements of cards on the reels. Subtract
the previous count of straight flushes with no Jokers:
10*4^5*5! - (10*4*5!) = 1224000
b) one Joker
As with straight flushes, there are 41 possibilities for non-Joker
cards' denominations. Assign suits to non-Joker cards arbitrarily,
but with five distinct cards we still have 5! arrangements of the
cards on the reels. Subtract our previous count of straight flushes
with one Joker:
41*4^4*5! - (41*4*5!) = 1239840
c) two Jokers
As with straight flushes, there are 64 possibilities for non-Joker
cards' denominations. Assign suits to non-Joker cards arbitrarily,
and with two interchangeable Jokers we have 5!/2 arrangements of the
cards on the reels. Subtract our previous count of straight flushes
with two Jokers:
64*4^3*(5!/2) - (64*4*5!/2) = 230400
Combining these cases gives the total outcomes counted as straights:
1224000 + 1239840 + 230400 = 2694240
>>>>>>>>>>>>>>>>>>>> END of Analysis >>>>>>>>>>>>>>>>>>>>>>>>>>
SUMMARY OF RESULTS
# of Jokers
Outcome Type
0 1 2 3 4 5 ALL
------------------------------------------------------------------
1. Five Jokers 0 0 0 0 0 1 1
2. 5 of kind/S 52 260 520 520 260 - 1612
3. Strgt Flush 4800 19680 15360 3680 - - 43520
4. 5 of kind/U 13260 16380 7800 1560 - - 39000
5. 4 of kind/S 12480 49920 74880 21280 - - 158560
6. 4 of kind/U 786240 748800 224640 - - - 1759680
7. Full House 1597440 599040 0 - - - 2196480
8. Flush 1470960 529440 53280 - - - 2053680
9. Straight 1224000 1239840 230400 - - - 2694240 |
