Formula for calculating Exotic wagers such as Trifecta and Superfecta - algorithm

Question

I am trying to create an application that will calculate the cost of exotic parimutuel wager costs. I have found several for certain types of bets but never one that solves all the scenarios for a single bet type. If I could find an algorithm that could calculate all the possible combinations I could use that formula to solve my other problems.
Additional information:
I need to calculate the permutations of groups of numbers. For instance;
Group 1 = 1,2,3
Group 2 = 2,3,4
Group 3 = 3,4,5
What are all the possible permutation for these 3 groups of numbers taking 1 number from each group per permutation. No repeats per permutation, meaning a number can not appear in more that 1 position. So 2,4,3 is valid but 2,4,4 is not valid.
Thanks for all the help.

Answer 1

Like most interesting problems, your question has several solutions. The algorithm that I wrote (below) is the simplest thing that came to mind.
I found it easiest to think of the problem like a tree-search: The first group, the root, has a child for each number it contains, where each child is the second group. The second group has a third-group child for each number it contains, the third group has a fourth-group child for each number it contains, etc. All you have to do is find all valid paths from the root to leaves.
However, for many groups with lots of numbers this approach will prove to be slow without any heuristics. One thing you could do is sort the list of groups by group-size, smallest group first. That would be a fail-fast approach that would, in general, discover that a permutation isn't valid sooner than later. Look-ahead, arc-consistency, and backtracking are other things you might want to think about. [Sorry, I can only include one link because it's my first post, but you can find these things on Wikipedia.]
## Algorithm written in Python ##
## CodePad.org has a Python interpreter
Group1 = [1,2,3] ## Within itself, each group must be composed of unique numbers
Group2 = [2,3,4]
Group3 = [3,4,5]
Groups = [Group1,Group2,Group3] ## Must contain at least one Group
Permutations = [] ## List of valid permutations
def getPermutations(group, permSoFar, nextGroupIndex):
for num in group:
nextPermSoFar = list(permSoFar) ## Make a copy of the permSoFar list
## Only proceed if num isn't a repeat in nextPermSoFar
if nextPermSoFar.count(num) == 0:
nextPermSoFar.append(num) ## Add num to this copy of nextPermSoFar
if nextGroupIndex != len(Groups): ## Call next group if there is one...
getPermutations(Groups[nextGroupIndex], nextPermSoFar, nextGroupIndex + 1)
else: ## ...or add the valid permutation to the list of permutations
Permutations.append(nextPermSoFar)
## Call getPermutations with:
## * the first group from the list of Groups
## * an empty list
## * the index of the second group
getPermutations(Groups[0], [], 1)
## print results of getPermutations
print 'There are', len(Permutations), 'valid permutations:'
print Permutations

Answer 2

This is the simplest general formula I know for trifectas.
A=the number of selections you have for first; B=number of selections for second; C=number of selections for third; AB=number of selections you have in both first and second; AC=no. for both first and third; BC=no. for both 2nd and 3rd; and ABC=the no. of selections for all of 1st,2nd, and third.
the formula is
(AxBxC)-(ABxC)-(ACxB)-(BCxA)+(2xABC)
So, for your example ::
Group 1 = 1,2,3
Group 2 = 2,3,4
Group 3 = 3,4,5
the solution is:: (3x3x3)-(2x3)-(1x3)-(2x3)+(2x1)=14. Hope that helps
There might be an easier method that I am not aware of. Now does anyone know a general formula for First4?

Answer 3

Revised after a few years:-
I re logged into my SE account after a while and noticed this question, and realised what I'd written didn't even answer you:-
Here is some python code
import itertools
def explode(value, unique):
legs = [ leg.split(',') for leg in value.split('/') ]
if unique:
return [ tuple(ea) for ea in itertools.product(*legs) if len(ea) == len(set(ea)) ]
else:
return [ tuple(ea) for ea in itertools.product(*legs) ]
calling explode works on the basis that each leg is separated by a /, and each position by a ,
for your trifecta calculation you can work it out by the following:-
result = explode('1,2,3/2,3,4/3,4,5', True)
stake = 2.0
cost = stake * len(result)
print cost
for a superfecta
result = explode('1,2,3/2,4,5/1,3,6,9/2,3,7,9', True)
stake = 2.0
cost = stake * len(result)
print cost
for a pick4 (Set Unique to False)
result = explode('1,2,3/2,4,5/3,9/2,3,4', False)
stake = 2.0
cost = stake * len(result)
print cost
Hope that helps

Answer 4

AS a punter I can tell you there is a much simpler way:
For a trifecta, you need 3 combinations. Say there are 8 runners, the total number of possible permutations is 8 (total runners)* 7 (remaining runners after the winner omitted)* 6 (remaining runners after the winner and 2nd omitted) = 336
For an exacta (with 8 runners) 8 * 7 = 56
Quinellas are an exception, as you only need to take each bet once as 1/2 pays as well as 2/1 so the answer is 8*7/2 = 28
Simple

Answer 5

The answer supplied by luskin is correct for trifectas. He posed another question I needed to solve regarding First4. I looked everywhere but could not find a formula. I did however find a simple way to determine the number of unique permutations, using nested loops to exclude repeated sequences.
Public Function fnFirst4PermCount(arFirst, arSecond, arThird, arFourth) As Integer
Dim intCountFirst As Integer
Dim intCountSecond As Integer
Dim intCountThird As Integer
Dim intCountFourth As Integer
Dim intBetCount As Integer
'Dim arFirst(3) As Integer
'Dim arSecond(3) As Integer
'Dim arThird(3) As Integer
'Dim arFourth(3) As Integer
'arFirst(0) = 1
'arFirst(1) = 2
'arFirst(2) = 3
'arFirst(3) = 4
'
'arSecond(0) = 1
'arSecond(1) = 2
'arSecond(2) = 3
'arSecond(3) = 4
'
'arThird(0) = 1
'arThird(1) = 2
'arThird(2) = 3
'arThird(3) = 4
'
'arFourth(0) = 1
'arFourth(1) = 2
'arFourth(2) = 3
'arFourth(3) = 4
intBetCount = 0
For intCountFirst = 0 To UBound(arFirst)
For intCountSecond = 0 To UBound(arSecond)
For intCountThird = 0 To UBound(arThird)
For intCountFourth = 0 To UBound(arFourth)
If (arFirst(intCountFirst) <> arSecond(intCountSecond)) And (arFirst(intCountFirst) <> arThird(intCountThird)) And (arFirst(intCountFirst) <> arFourth(intCountFourth)) Then
If (arSecond(intCountSecond) <> arThird(intCountThird)) And (arSecond(intCountSecond) <> arFourth(intCountFourth)) Then
If (arThird(intCountThird) <> arFourth(intCountFourth)) Then
' Debug.Print "First " & arFirst(intCountFirst), " Second " & arSecond(intCountSecond), "Third " & arThird(intCountThird), " Fourth " & arFourth(intCountFourth)
intBetCount = intBetCount + 1
End If
End If
End If
Next intCountFourth
Next intCountThird
Next intCountSecond
Next intCountFirst
fnFirst4PermCount = intBetCount
End Function
this function takes four string arrays for each position. I left in test code (commented out) so you can see how it works for 1/2/3/4 for each of the four positions