You're missing some terms in the third equality: the sum disappeared and you've dropped the terms where you take the commutators with the second spin operator in each product.
a) That's correct. b) The result is nonzero if $l+1=n$. Hint: first work out the commutators between $S^+_i, S^-_j, S^z_k$ and then do the computation more carefully. c) What is the eigenvalue of $S^z_i$ on $\vert \Omega\rangle$?
Jules Lamers
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