An adsorbing filter allows gas particles to stick to locations
on the filter surface. Once a particle sticks to a location, that
location is filled. The filter can no longer remove gas particles
when all locations are filled. Each 1.0 nm 2 of the filter surface
has six adsorbing locations, each capable of adsorbing one gas
particle. In the volume just adjacent to one 1.0 nm 2 area, there
are 100 gas particles. Each of these particles has a slightly dif-
ferent energy, making each particle unique. In how many ways
can these 100 particles adsorb onto the six locations?
The Attempt at a Solution
Consider first 6 particles. You can put the first particle in one of 6 boxes, the second particle in one of 5 remaining boxes, etc., which gives 6! ways to put 6 particles into 6 boxes.
Consider 7 particles. The excluded particle can be thought of as being in the "excluded" box, so you get 7! ways to put 7 distinguishable particles into 6 boxes.
Consider 8 particles. You still have 6! ways to put 6 particles in the 6 boxes. Multiply this by the number of ways can you can put 2 particles out of 8 into the "excluded" box. This latter number is 8 X 7, so the total number of ways to put 8 distinguishable particles into 6 boxes is
6! X 7 X 8 = 8!.
For 9 particles, I think you can have 7 X 8 X 9 ways to put 3 distinguishable particles out of 9 into the "excluded" box, so the total number of ways to put 9 distinguishable particles into 6 boxes is
6! X 7 X 8 X9 = 9!
Following this logic, the total number of ways to put 100 distinguishable particles into 6 boxes is 100!.
Is this correct? Or am I missing something?