  We perform a theoretical study on the $D\bar{D}K$ three body system, using
the fixed center approximation to the Faddeev equations, considering the
interaction between $D$ and $K$, $D$ and $\bar{D}$ from the chiral unitary
approach. We assume the scattering of $K$ meson on a clusterized system
$D\bar{D}$, where a scalar meson $X(3720)$ could be formed. Thanks to the
strong $DK$ interaction, where the scalar $D^*_{s0}(2317)$ meson is dynamically
generated, a resonance structure shows up in the modulus squared of the three
body $K$-$(D\bar{D})_{X(3720)}$ scattering amplitude and supports that a
$D\bar{D}K$ bound state can be formed. The result is in agreement with previous
theoretical studies, which claim a new excited hidden charm $K$ meson,
$K_{c\bar{c}}(4180)$ with quantum numbers $I(J^P) = \frac{1}{2}(0^-)$ and mass
about $4180$ MeV. It is expected that these theoretical results motivate its
search in experimental measurements.
