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We present deep galaxy counts in the $K$ ($\lambda 2.2 \mu$m) band, obtained at the W.M.~Keck 10-m telescope. The data reach limiting magnitudes $K \sim 24^m$, about five times deeper than the deepest published $K$-band images to date. The counts are performed in five small ($\sim 1$ arcmin), widely separated high-latitude fields. Extensive Monte-Carlo tests were used to derive the completeness corrections and minimize photometric biases. The analysis of the first three fields shows that the counts continue to rise, with no sign of a turnover, down to the limits of our data, with the logarithmic slope of $d \log N / d m = 0.315 \pm 0.02$ between $K = 20^m$ and $K = 24^m$. This implies a cumulative surface density of $\sim 5 \times 10^5$ galaxies/degree$^2$, or $\sim 2 \times 10^{10}$ over the entire sky, down to $K = 24^m$. Our counts are in good agreement with, although slightly lower than, those from the Hawaii Deep Survey by Cowie and collaborators; the discrepancies may be due to the small differences in the aperture corrections. The observed field-to-field variations are as expected from the Poissonian noise and galaxy clustering as described by the angular two-point correlation function for faint galaxies. We compare our counts with some of the available theoretical predictions. The data do not require models with a high value of $\Omega_0$, but can be fit well by models with no (or little) evolution, and cosmologies with a $low$ value of $\Omega_0$. Given the uncertainties in the models, it may be premature to put useful constrains on the value of $\Omega_0$ from the counts alone. Optical-to-IR colors are computed, using CCD data obtained previously at Palomar. We find a few red galaxies with $(r-K) \geq 5^m$, or $(i-K) \geq 5^m$; these may be ellipticals at $z \sim 1$. While the redshift distribution of galaxies in our counts is still unknown, the flux limits reached would allow us to detect unobscurred $L_*$ galaxies out to substantial redshifts ($z > 3$?).
