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Expert: - 11/16/2009


Question
Consider the following variables:
                      
                      Z
              A1      B2       C3
            A4  D1   B5  E2   C6  F3
            G4  D7   H5  E8   I6  F9

How many groups can be made with the variables from the set if two
variables with the same number or same letter cannot be in the same set and Z cannot be with A1, B2, or C3 (include the empty set)

Now I think I got the answer. Can you just check my work to make sure its right?

Okay. First I split the variables up like so:
             GROUP 1   GROUP 2  GROUP 3   
              A1     |   B2    |   C3
            A4 D1    |  B5 E2  |  C6 F3
            G4 D7    |  H5 E8  |  I6 F9

I ignored Z for the moment. My motivation for splitting the variables up like that was because any variable from GROUP 1 wont affect GROUP 2 or GROUP 3, so we can find the total number of groups in just one group and cube it since the same number of groups will be in every other group.

If we do GROUP 1, simple listing gives the groups

{empty set}
A1
D1
G4
D7
A4
A1 G4
A1 D7
A4 D1
A4 D7
D1 G4
G4 D7
A1 G4 D7

For a total of 13 (THOUGH I PROBABLY MISCOUNTED-CAN YOU VERIFY THIS?).
By symmetry, there will then be 13 groups in GROUP 2 and 13 groups in GROUP3, for a total of 2197.

But that is not including Z in any groups. Z cannot be with A1, B2, or C3, and we see that there are 4 groups in GROUP 1 that include A1, so by symmetry there will be 4 groups in GROUP 2 that include B2, and four groups in GROUP 3 that include C3. Hence, there are only (13-4)^3=729 groups which can have the Z added to it.

So the total number of groups is 13^3+9^3=2926



Can you PLEASE tell me if there is a mistake in there somewhere?


Answer
The counting looked good, and, well, I looked over it,
and the only eror I think I saw was when the sets were counted.
It's easiest to group them in groups of five.

A1
D1
G4
D7
A4  5

A1 G4
A1 D7
A4 D1
A4 D7
D1 G4  10

G4 D7
A1 G4 D7  12

Yeah, that's 12 and not 13.

This make the total number not 13^3, but 12^3.  Now 12^3 = 1,728.
This makes 12-4=8, and 8^3=512.

Now when you add 12^3 + 8^3 you get 1,728 + 512 = 2,240.

So 2,926 was in the ball park, but not right.

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