Abstract We used very deep near-infrared imaging data taken with the Multi-Object InfraRed Camera and Spectrograph (MOIRCS) on the Subaru Telescope to investigate the number counts of Distant Red Galaxies (DRGs). We observed a $4' \times 7'$ field in the Great Observatories Origins Deep Survey-North (GOODS-N), and our data reached $J=24.6$ and $K=23.2$ ($5\,\sigma$, Vega magnitude). The surface density of DRGs selected by $J-K \gt 2.3 J-K \gt 2.3$ is $2.35 \pm 0.31 \,\mathrm{arcmin}^{-2}$ at $K < 22$ and $3.54 \pm 0.38 \,\mathrm{arcmin}^{-2}$ at $K < 23$, respectively. These values are consistent with those in the GOODS-South and FIRES. Our deep and wide data suggest that the number counts of DRGs turn over at $K \sim 22$, and the surface density of the faint DRGs with $K \gt 22$ is smaller than that expected from the number counts at the brighter magnitude. The result indicates that while there are many bright galaxies at $2 < z < 4$ with the relatively old stellar population and/or heavy dust extinction, the number of faint galaxies with a similar red color is relatively small. Different behavior patterns of the number counts of the DRGs and bluer galaxies with $2 < z_{\mathrm{phot}} < 4$ at $K \gt 22$ suggest that the mass-dependent color distribution, where most of the low-mass galaxies are blue, while more massive galaxies tend to have redder colors, had already been established at that epoch.