Combinatorics - 2D backpack
Suppose that we want to distribute 2 identical black balls and 3 identical white balls into 4 distinct bins, so that each bin has at least one ball.
How many ways can we do this?
Bonus: Why the title "2D backpack" ?
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1 solution
3 bins must have 1 ball, and 1 must have 2.
There are 3 possible combinations for the bin with 2 balls:
WW
BW
BB
If the combination is BB, then there are 3 bins that the other black ball can be in.
If the combination is BW, then there are 3 bins that the other white ball can be in.
If the combination is WW, then there is only 1 possibility for the rest of the bins: They all have a black ball.
3 + 3 + 1 = 7
Since there are 4 choices for the bin with 2 balls, we must then multiply by 4.
7 × 4 = 2 8