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

bkraabel

## 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
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?
thanks

## Answers and Replies

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.

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

bkraabel
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

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.

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.

Mentor
Yes. This is the correct answer.

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