I've been spinning my wheels on this for an embarrassing amount of time. This is not for an assignment, I'm just studying.
Three balls are placed at random in
three boxes. There is no restriction
on the number of balls per box; list
the 27 possible outcomes of this
experiment.
I could only come up with 10 where order of placing the balls into the boxes doesn't matter.
{(0,0,3), (0,1,2), (0,2,1), (0,3,0), (1,0,2), (1,1,1), (1,2,0), (2,0,1), (2,1,0), (3,0,0)}
What am I doing wrong?
Best Answer
Using the stars-and-bars technique, for your interpretation of the problem, there are in fact ${3+3-1\choose 3}=10$ possible outcomes.
As Qiaochu Yuan said in his comment, 27 is the result when the three balls are distinguishable (and I agree with him that it is not clear in the problem statement)—for each of the 3 balls, there are 3 possible locations, so $3^3=27$ possible outcomes.