  A Dicke state and its decohered state are invariant for permutation. However,
when another qubits state to each of them is attached, the whole state is not
invariant for permutation, and has a certain asymmetry for permutation. The
amount of asymmetry can be measured by the number of distinguishable states
under the group action or the mutual information. This paper investigates how
the coherence of a Dicke state affects the amount of asymmetry. To address this
problem asymptotically, we introduce a new type of central limit theorem by
using several formulas on hypergeometric functions. We reveal that the amount
of the asymmetry in the case with a Dicke state has a strictly larger order
than that with the decohered state in a specific type of the limit.
