good point,
yes, let me take a look at solution.
- it is about
n definitively named marbles eg: marbel_a, marbel_b, ....
- it is about no-named colors meaning that coloring induce a partition not a surjection
- boxes are also no-named
- there are no empty boxes
- there could be
k common colors among the box-colors and the marble-colors that make these
k common colors to be special.
Example : two marbles, "red" and"blue" in a "blue" box is a different situation to the same marbles in a "yellow" box. "Blue" is a special color, but it is still no-named (of course, if "Blue" is the only one special color, one could name it without any risk.)
The formula required depends only on
n and
k, summing all distinct possibilities of:
- coloring the named marbles with no-name colors (partitioning them)
- placing them in the no-name boxes
- coloring the boxes
for all possible amounts of boxes and colors.
=============
In fact I had some difficulties with the common colors.
Suppose there are four boxes and four marbles, and I am colorblind. Then I ask someone to put for me the marbles in the right boxes. I find that there are four different common colors and they are now one-to-one at their places.
Of course, the next thing I do is to mark them with a permanent marker Does not this means that:
- establishing a bijection between boxes colors and marbles colors is equivalent to
- giving names to the no-named colors ?
So, in my solution I considered that the common colors are named colors. That is why I asked here for another solution, to clarify this philosophy of colors.