I'm betting on sports from time to time, mainly football (soccer) and I usually bet very small amounts of money on the "correct score". I usually make a combination of 3 games so I get the odds up. Of course since it's extremely hard to get 3 scores correct in the one and same combination I loose almost every time but it's fine since it's such a small amount of money. I bet like 0.10€ per combination.
But I have always wondered one thing and since my math skills are extremely limited I need help with this.
So here's the question:
How many different full time score-combinations can it possibly be out of a 3 game-combination in football? If you follow some of these rules…
Rules:
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In this calculation you will not count any away team wins. Meaning the only scores possible are home team wins and ties.
-
There can only be a maximum of 1 tie score per combination.
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The only scores that should be a part of this calculation are: 0-0, 1-0, 1-1, 2-0, 2-1, 2-2, 3-0, 3-1, 3-2, 4-0, 4-1.
Even with these rules I obviously understand that it will be a lot of different combinations, probably over 200 but I'd still like to know if anyone would like to help.
Example:
We have 3 games. Of course it's the same 3 games that we bet on in all these correct score – combinations.
(G1 = Game 1, G2 = Game 2, G3 = Game 3).
Combo 1. G1: 1-0 G2: 1-0 G3: 1-0
Combo 2. G1: 1-0 G2: 1-0 G3: 0-0
Combo 3. G1: 1-0 G2: 0-0 G3: 1-0
Combo 4. G1: 0-0 G2: 1-0 G3: 1-0
Combo 5. G1: 1-0 G2: 1-0 G3: 1-1
Combo 6. G1: 1-0 G2: 1-1 G3: 1-0
Combo 7. G1: 1-1 G2: 1-0 G3: 1-0
Combo 8. G1: 2-0 G2: 2-0 G3: 2-0
Combo 9. G1: 2-0 G2: 2-0 G3: 1-0
Combo10. G1: 2-0 G2: 1-0 G3: 1-0
Combo11. G1: 2-0 G2: 1-0 G3: 2-0
Combo12. G1: 1-0 G2: 1-0 G3: 2-0
Combo13. G1: 1-0 G2: 2-0 G3: 2-0
Combo14. G1: 2-0 G2: 2-0 G3: 0-0
Combo15. G1: 2-0 G2: 0-0 G3: 2-0
Combo16. G1: 0-0 G2: 2-0 G3: 2-0
Combo17. G1: 2-0 G2: 0-0 G3: 1-0
Combo18. G1: 2-0 G2: 1-0 G3: 0-0
Combo19. G1: 0-0 G2: 1-0 G3: 2-0
And so on…
As you can see I don't really have a good counting system for solving this and it gets pretty messy and not very easy to follow. But I hope some of you guys understand the question and what I'm looking for.
I want to know ALL different scores possible, in any order, for a 3-game combination following the rules and only using the scores mentioned in the rules.
I don't want to give away what I'm supposed to do with this calculation if someone can provide it but I promise, it's not what you probably think.
I would be very satisfied if someone just could give the exact number of possible score-combinations for 3 games, it's not necessary to demonstrate ALL the possible scores. Wouldn't mind if someone did that too though hehe… 🙂
Thanks for reading all of this and I hope I can get an answer on this! Take care
Best Answer
OK, given that there are $3$ possible tie outcomes, and $8$ non-tie outcomes, you have:
Number of games without any ties: $8*8*8= 512$
Number of games with $1$ tie: $3*8*8*3$ (two games no tie, and one game a tie, and that tie game can be the first, second, or third game) = $576$
Total: $1088$
And you're right, I'm not going to show all $1088$! :)
But if you were to do this, you should do it a bit more systematically than the way you do. And, doing it more systematically, should also reveal why the number is what it is.
First, let's do the no ties:
1-0 1-0 1-0
1-0 1-0 2-0
1-0 1-0 2-1 ...
1-0 1-0 4-1
Now change the second game to 2-0, and vary the third one:
1-0 2-0 1-0
1-0 2-0 2-0
1-0 2-0 2-1
...
change second one again, and do the next 8:
1-0 2-1 1-0
1-0 2-1 2-0
...
1-0 2-1 4-1
... I think you get the pattern now, so after we have gone through all 8 games for the second game, we eventually end up with
...
OK, so now change the first game, and start again with second and third:
2-0 1-0 1-0
2-0 1-0 2-0
...
so we get another 8*8:
2-0 4-1 4-1
2-1 1-0 1-0
Etc.
... until ...
Now let's do the combination with 1 tie:
...
...
0-0 4-1 4-1
1-1 1-0 1-0
...
1-1 4-1 4-1
2-2 1-0 1-0
...
OK, but second game could be a tie instead of the first, so:
...
4-1 0-0 4-1
1-0 1-1 1-0
...
and now the third game a tie ...
...