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

• bkraabel
In summary: The problem statement says that each box can accommodate only one particle. 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?thanksNo, the problem statement is asking for the number of 6-element subsets (where the order of the 6 elements doesn't matter) in a set of 100 distinguishable elements.

## 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

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

bkraabel said:
In how many ways can these 100 particles adsorb onto the six locations?

bkraabel said:
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.

bkraabel said:
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.

Chestermiller said:
Yes. This is the correct answer.

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

## Question 1: What is the total number of possible ways to arrange 100 distinguishable particles into 6 boxes?

The total number of ways to arrange 100 distinguishable particles into 6 boxes is equal to 6100 or approximately 1.578 x 1089. This is because for each of the 100 particles, there are 6 possible boxes it could be placed in, resulting in 6100 possible combinations.

## Question 2: Is there a formula or equation to calculate the number of ways to arrange 100 distinguishable particles into 6 boxes?

Yes, there is a formula known as the "stars and bars" formula that can be used to calculate the number of ways to arrange n distinguishable objects into k boxes. For this particular scenario, the formula would be (100+k-1) choose (k-1), resulting in 105 choose 5 or approximately 8.347 x 1014 possible combinations.

## Question 3: How does the number of ways to arrange 100 distinguishable particles into 6 boxes compare to the number of ways to arrange 100 indistinguishable particles into 6 boxes?

The number of ways to arrange 100 distinguishable particles into 6 boxes is significantly larger than the number of ways to arrange 100 indistinguishable particles into 6 boxes. This is because distinguishable particles can be differentiated and placed in any order, while indistinguishable particles are identical and therefore have fewer possible combinations.

## Question 4: Are there any constraints or restrictions on how the 100 distinguishable particles can be arranged into 6 boxes?

Yes, there are a few constraints that must be considered when arranging the particles into 6 boxes. First, all 100 particles must be placed into one of the 6 boxes. Second, no box can be left empty. And third, the order of the particles within each box does not matter.

## Question 5: Can any patterns or trends be observed in the number of ways to arrange 100 distinguishable particles into 6 boxes?

Yes, as the number of particles and boxes increases, the total number of possible combinations also increases exponentially. This is because for each additional particle, there are more possible boxes it could be placed in, resulting in a larger number of combinations. Additionally, if the number of particles is smaller than the number of boxes, the number of combinations will be significantly smaller due to the constraints mentioned above.