= Solution
{c}
Restrict $S$ to the <vector subspace> $W=\operatorname{im}T$. This restriction has <kernel of a linear map> $W\cap\ker S$ and <image of a linear map> $S(W)=\operatorname{im}(ST)$. The <rank-nullity theorem> on $W$ therefore gives
$$
r(ST)=r(T)-\dim(\operatorname{im}T\cap\ker S).
$$
Apply the <rank-nullity theorem> to $T$ and $ST$ on $V$ to obtain the exact <nullity of a composition> identity:
$$
n(ST)=n(T)+\dim(\operatorname{im}T\cap\ker S).
$$
Since the intersection is a <vector subspace> of $\ker S$, its <dimension> is at most $n(S)$. Hence
$$
\boxed{n(ST)\leq n(T)+n(S).}
$$
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