Distributing 6 Distinct Molecules Across 3 Energy Levels

[patcho]

Hi,

I'm not quite sure about this question:

'How many Ways can 6 distinguishable molecules be placed in 3 different energy levels with 3 molecules in the 1st level, 2 in the 2nd level and 1 in the 3rd level, ignoring energy required?'

If it was just how many ways to place them in 3 different levels it would be easy but how to always keep 3 molecules in the first, 2 in the 2nd and 1 in the first confuses me.

I know that the number of Ways is less than before and I'm thinking along the lines of having to divide the number obtained if it was just 3 different energy levels, by 3!2!1! (the number of molecules in each level). This gives: W=10 which I think is very wrong!

Any help appreciated!

 

[Galileo]

Take a slightly more general question:

How many ways can N numbered balls be grouped into three sets such that the first group contains $n_1$ balls, the second set $n_2$ balls and the third set $N-n_1-n_2$ balls?

Hint: How many ways are there to choose the first set? After that, in how many ways can you choose the second set?

 

[thepatientmental]

Hi there,

Thank you for your question. It seems like you are trying to figure out the number of possible arrangements for 6 distinguishable molecules in 3 different energy levels, with specific numbers of molecules in each level.

In this case, you can use the concept of combinations to find the number of ways to place the molecules. The formula for combinations is nCr = n! / (r! * (n-r)!), where n is the total number of objects and r is the number of objects being chosen.

In this problem, we have 6 molecules and we want to choose 3 for the first level, 2 for the second level, and 1 for the third level. So, the number of ways to place these molecules would be:

6C3 * 3C2 * 1C1 = (6! / (3! * (6-3)!) * (3! / (2! * (3-2)!) * (1! / (1! * (1-1)!)
= (6*5*4 / (3*2*1)) * (3*2 / (2*1)) * (1 / (1*0))
= 20 * 3 * 1
= 60

So, there are 60 ways to place 6 distinguishable molecules in 3 different energy levels with 3 molecules in the first level, 2 in the second level, and 1 in the third level.

I hope this helps clarify the concept for you. If you have any further questions, please feel free to ask. Best of luck with your studies!