For our project, we will only consider the pairing model. A simple space is the \( (1/2)^2 \) space with four single-particle states
| Index | \( n \) | \( l \) | \( s \) | \( m_s \) |
| 1 | 0 | 0 | 1/2 | -1/2 |
| 2 | 0 | 0 | 1/2 | 1/2 |
| 3 | 1 | 0 | 1/2 | -1/2 |
| 4 | 1 | 0 | 1/2 | 1/2 |
For \( N=2 \) there are 4 states with \( M=0 \); show this by hand and confirm your code reproduces it.