  Many-body states that are invariant under particle relabelling, the
permutation symmetric states, occur naturally when the system dynamics is
described by symmetric processes or collective spin operators. We derive
expressions for the reduced density matrix for arbitrary subsystem
decomposition for these states and study properties of permutation symmetric
states and their subsystems when the joint system is picked randomly and
uniformly. Thus defining a new random matrix ensemble, we find the average
linear entropy and von Neumann entropy which implies that random permutation
symmetric states are marginally entangled and as a consequence the tripartite
mutual information (TMI) is typically positive, preventing information from
being shared globally. Applying these results to the quantum kicked top viewed
as a multi-qubit system we find that entanglement, mutual information and TMI
all increase for large subsystems across the Ehrenfest or log-time and saturate
at the random state values if there is global chaos. During this time the
out-of-time order correlators (OTOC) evolve exponentially implying scrambling
in phase space. We discuss how positive TMI may coexist with such scrambling.
