Missing something pretty simple in counting energy levels?

[genericusrnme]

Homework Statement

Determine the no. of energy levels with different values of total spin for a system of N particles with spin 1/2

The Attempt at a Solution

Let f be the no. ways to get a z projection of spin, then

f(\sigma )=Binomial(N,\frac{1}{2}N + \sigma)
Where Binomial(a,b) is a chose b

That part makes perfect sense
To each energy level with a given S there corresponds 2S+1 states with σ=S...-S
Again, I know this
Hence it is easy to see that the no. different energy levels with a given value of S are

n(S)=f(s)-f(s+1)

And that is where it lost me, I don't understand why this is and I feel like I'm missing something simple..

Could anyone offer any help?

 

[vela]

How exactly does the Hamiltonian depend on spin here? It seems you'd have to know that to be able to count the number of energy levels.

 

[genericusrnme]

How exactly does the Hamiltonian depend on spin here? It seems you'd have to know that to be able to count the number of energy levels.

There's no mention of the hamiltonian at all.

What I quoted in the problem statement was everything that I'm given in the problem and, unless I've missed something, there has been no mention of any assumptions being made about the hamiltionian up to this point.

It's at the end of a section that told of how to use youngs diagrams and symmetries of the coordinate and spin wavefunctions under permutations so I'm guessing I have to use that but I'm really not following.