3 x 3 colors ...
Consider a grid sized 3 x 3. Using three colors, red, white , and blue, (to color the cells in the grid) ... how many distinct patterns can be displayed if all three colors must appear in every column and every row?
posted by Zaux at
8:19 AM
7 Comments:
Without thinking too hard, I'll say 18
ABC
DEF
GHI
A has 3 choices
B,D have two choices
CG are fixed 1 choice
E has 1 choice if BD are different
E has 2 choices if BD are the same
each occurs 1/2 of the time
i.e. if A is W then BD is RR,RB,BR,BB, same 2/4
FHI are then fixed at 1
so
3*2*2*(0.5*2+0.5*1)=18
Answer
18
Cam
Hi Cam ...
one of the few times you did not present the correct answer.
Meh, looks like E never really has a choice
ABC
DEF
GHI
A has 3 choices
B,D have two choices
CG are fixed 1 choice
E has 1 choice regardless if BD are different or the same
FHI are then fixed at 1
so
3*2*2*1=12
Answer
12
looking for distinctly different ... rotations of arrangements are not distinctly different
12/4=3 for truly distinct. No rotations/mirror images.
Cam
Explanation for why E has 1 choice regardless if BD are different or the same
Only 3 colors
if B,D are different 2 of the 3 colors are taken, and E must choose the remaining 1
if B,D are the same and E is chosen as the same as A then the result is invalid
Why? If E=A
C must be different from A,B, so C is the color not chosen
F must be different from D and E. but D=B,E=A thus F=C which puts two of the same color in the same column, thus invalid.
Thus only 1 valid choice is actually available for E.
Cam
Hi Cam ....
3 is correct ... but you already knew that :-)
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