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_capprox$3.54, classifying this material as a weakly-coupled superconductors.
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