  A detailed zero-field and transverse-field muon spin relaxation/rotation
($\mu$SR) experiemnts have been carried out on the recently discovered
non-centrosymmetric superconductor W$_3$Al$_2$C to speculate about its
superconducting ground state. Bulk nature of superconductivity below 7.6 K is
confirmed through magnetization measurements. No change in the $\mu$SR spectra
collected above and below $T_c$ is visible, ruling out the possibility of
spontaneous magnetic field below $T_c$. This confirms that time-reversal
symmetry is preserved for W$_3$Al$_2$C upon entering in the superconducting
ground state. Temperature dependent superfluid density [$\rho_s(T)$], which
directly reflects the superconducting gap symmetry is obtained by the analysis
of spectra obtained from the transverse-field $\mu$SR experiments. Despite a
non-centrosymmetric structure, W$_3$Al$_2$C adopts a fully gaped spin-singlet
superconducting ground state with a zero temperature value of gap $\Delta_0$ =
1.158(8) meV with gap-to-$T_c$ ratio 2$\Delta_0/k_BT_c\approx$3.54, classifying
this material as a weakly-coupled superconductors.
