Help needed resolving Possible Outcomes.

[sugarkayne]

How many possible outcomes when 3 football matches are played keeping in mind that there could be a draw, win or loss for any of the teams?

An example is as follows:

Teams HOME DRAW AWAY

Chelsea vs Arsenal

Liverpool Vs Manchester United

Real Madrid Vs Ac Milan


Intepretation of the above statement:

HOME - Means Chelsea, Liverpool, Real Madrid, Win their matches.
DRAW - Means the matches end in a draw
AWAY - Means Arsenal, Manchester United, Ac Milan, win their matches.

WOuld like to have answers in the following format.
HHH,DDD,AAA,HAD,DHA,DDA,AAD,HDA and so on and so on etc.....

Thanks

[jemidiah]

Each match's outcome is independent of the others. Each match has 3 outcomes, home team wins, home team loses, or draw. The rule of product says that there are 3*3*3 = 27 outcomes overall. The answers can be explicitly found by counting in base 3. I'll give an example in base 10, since it's more familiar.

Suppose instead of just home win-home loss-draw, each match had 10 outcomes--say, (0) draw, (1) home win by 3 or fewer points, (2) home win by 7 or more points, (3) home loss by 3 or fewer points, etc. Continue numbering them 0, 1, 2, ..., 9. Any overall outcome can be described using three digits in order--for instance, 230 would mean the first game had outcome 2 [home win by 7 or more], the second game had outcome 3 [home loss by 3 or fewer], and the third game had outcome 0 [draw]. It's easy to see that every number from 000 to 999 represents a different outcome, and that all outcomes are of this form. Since there are 1000 numbers from 000 = 0 to 999, there are 1000 = 10*10*10 outcomes.

The case above is basically identical, but instead of numbers 0-9, you limit yourself to numbers 0-2. Say 0 = home win, 1=home loss, 2=draw, so 212 means the first game was a draw, the second game was a home loss, and the third game was another draw. Just like there are 10*10*10 3-digit numbers in base 10, there are 3*3*3 3-digit numbers in base 3. So, again, there are 27 outcomes.

Explicitly...

000
001
002
010
011
012
020
021
022
100
101
102
110
111
112
120
121
122
200
201
202
210
211
212
220
221
222

are all outcomes.