Let's take a specific example: \( \Omega = 3 \) so there are 6 single-particle states, and \( N = 3 \), with \( v= 1,3 \). Therefore there are two distinct eigenvalues, $$ E = -2G, 0 $$ Now let's work this out explicitly. The single particle degrees of freedom are defined as
| Index | \( k \) | \( m \) |
| 1 | 1 | -1/2 |
| 2 | -1 | 1/2 |
| 3 | 2 | -1/2 |
| 4 | -2 | 1/2 |
| 5 | 3 | -1/2 |
| 6 | -3 | 1/2 |
There are \( \left( \begin{array}{c}6 \\ 3 \end{array} \right) = 20 \) three-particle states, but there are 9 states with \( M = +1/2 \), namely \( | 1,2,3 \rangle, |1,2,5\rangle, | 1,4,6 \rangle, | 2,3,4 \rangle, |2,3,6 \rangle, | 2,4,5 \rangle, | 2, 5, 6 \rangle, |3,4,6 \rangle, | 4,5,6 \rangle \).