Probabilities of molecules being somewhere

[adamc637]

A box is separated by a partition into two parts of equal volume. The left side of the box contains 500 molecules of nitrogen gas; the right side contains 100 molecules of oxygen gas. The two gases are at the same temperature. The partition is punctured, and equilibrium is eventually attained. Assume that the volume of the box is large enough for each gas to undergo a free expansion and not change temperature.

What is the probability that the molecules will be found in the same distribution as they were before the partition was punctured, that is, 500 nitrogen molecules in the left half and 100 oxygen molecules in the right half?

I have no idea how to start this. Would i do

(500!*100!)/600!

?

 

[thepatientmental]

Yes, that is the correct approach. To find the probability of the molecules being in the same distribution as before, we need to calculate the number of ways the molecules can be arranged in the two halves of the box. This can be done using the formula for permutations, which is n!/r!, where n is the total number of molecules and r is the number of molecules in one half of the box. In this case, n=600 and r=500 for nitrogen and r=100 for oxygen. So the probability would be (500!*100!)/600!, which is approximately 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000