= Solution
Total symmetry makes a component depend only on its number $n_\uparrow$ of up indices. The possible values are $0,1,\ldots,N$, giving
$$
\boxed{\dim\operatorname{Sym}^N(\mathbb C^2)=N+1=2s+1}.
$$
A normalized basis is $|n_\uparrow,N-n_\uparrow\rangle$, with $S_z$ <eigenvalue> $\hbar(n_\uparrow-N/2)$. It is precisely one spin-$N/2$ multiplet. These are independent complex amplitudes before normalization and removal of a common <quantum phase>, rather than $N+1$ independent real physical parameters.
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