 Solution

Use the computational basis ordered as $|000⟩,|001⟩,|010⟩,|011⟩,|100⟩,|101⟩,|110⟩,|111⟩$. Since $Z|0⟩=|0⟩$ and $Z|1⟩=-|1⟩$,For arbitrary $N$, a computational basis vector with $q$ ones has $N-q$ contributions $+1$ and $q$ contributions $-1$, giving eigenvalue $N-2q$. All $q=0,\dots ,N$ occur, soThe binomial coefficient counts which $q$ sites are excited. Thus there are exactly $N+1$ distinct magnetization sectors of a spin chain, and their dimensions sum to ${\sum }_q\left(\genfrac{}{}{0ex}{}{N}{q}\right)=2^N$, the full Hilbert-space dimension.