  Ever since the famous Erd\H{o}s-Ko-Rado theorem initiated the study of
intersecting families of subsets, extremal problems regarding intersecting
properties of families of various combinatorial objects have been extensively
investigated. Among them, studies about families of subsets, vector spaces and
permutations are of particular concerns.
  Recently, the authors proposed a new quantitative intersection problem for
families of subsets: For $\mathcal{F}\subseteq {[n]\choose k}$, define its
\emph{total intersection number} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in
\mathcal{F}}|F_1\cap F_2|$. Then, what is the structure of $\mathcal{F}$ when
it has the maximal total intersection number among all families in ${[n]\choose
k}$ with the same family size? In \cite{KG2020}, the authors studied this
problem and characterized extremal structures of families maximizing the total
intersection number of given sizes.
  In this paper, we consider the analogues of this problem for families of
vector spaces and permutations. For certain ranges of family size, we provide
structural characterizations for both families of subspaces and families of
permutations having maximal total intersection numbers. To some extent, these
results determine the unique structure of the optimal family for some certain
values of $|\mathcal{F}|$ and characterize the relation between having maximal
total intersection number and being intersecting. Besides, we also show several
upper bounds on the total intersection numbers for both families of subspaces
and families of permutations of given sizes.
