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**1. Homework Statement**

Okay, so the problem asks to show that if you have N number of O2 molecules with spin either 1, 0, or -1, the most probable value for either of those is N/3, assuming they don't interact at all and no B field, so basically 3-sided coins.

**2. Homework Equations**

Asked to start with the multinomial coefficient formula:

http://en.wikipedia.org/wiki/Multinomial_theorem

And Stirling's crude approximation for N! = N*ln(N) - N

**3. The Attempt at a Solution**

So Probability = [tex]\frac{\Omega}{\Omega_t}[/tex], right?

In my case [tex]\Omega_t = 3^N[/tex], so I substituted that. And for [tex]\Omega[/tex] I substituted the multinomial expansion coefficient formula, so I got my probability. Then I simplified the factorial on top and the ones in Product function on the bottom of the fraction and then multiplied them out, since I'm only dealing with 3 choices which I label x,y,z for clarity, so they are easy to multiply out.

Now I'm stuck. I want to take the derivative of this and set it to 0, but I'm not sure what to take the derivative with respect to. N? Also, I'm still stuck with an x, y, and z, but I'm thinking I can set those equal to 1.

[tex](x+y+z)^N = 3^N => x=y=z=1[/tex]

But that also seems dubious to me.

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