# How many ways to put 100 distinguishable particles into 6 boxes?

## Homework Statement

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
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?
thanks

Chestermiller
Mentor
The problem statement says that each box can accommodate only one particle. This problem requires you to find the number of combinations of 100 items taken 6 at a time.

Not to sound redundant, but the key word here is: combinations

So if I get your drift, then the number of 6-element subsets (where the order of the 6 elements doesn't matter) in a set of 100 distinguishable elements would be 100 choose 6, which is 1.2 X 10^9. Is that the idea?
thanks

DrClaude
Mentor
In how many ways can these 100 particles adsorb onto the six locations?

So if I get your drift, then the number of 6-element subsets (where the order of the 6 elements doesn't matter) in a set of 100 distinguishable elements would be 100 choose 6, which is 1.2 X 10^9. Is that the idea?
thanks

I have a different on the question. For me, the use of "how many ways" implies that it matters which particle is where on the filter.

Chestermiller
Mentor
So if I get your drift, then the number of 6-element subsets (where the order of the 6 elements doesn't matter) in a set of 100 distinguishable elements would be 100 choose 6, which is 1.2 X 10^9. Is that the idea?
thanks
Yes. This is the correct answer.

DrClaude
Mentor
Yes. This is the correct answer.

As I said above, I disagree. I don't think this is what the problem is asking.