A Color Gamut refers to colors within a Color Space which is a representation of which colors exist within it and which ones are not.

It does not usually define a representation which is why there is not specific number of colord within it. sRGB for example use 3 chromacities which form a triangle within all possible colors. When one uses an 8-bit-per-component or 24 bits-per-pixel, it has 16,777,216 colors. When one uses 10-bits per color component or 30 bits-per-pixel, it has 1,073,741,824. TIFF files can use 32-bit values per component and so are able to represent even more colors with the sRGB color space.

So, there is no answer to your question but the screen determines how many colors can be displayed. Even so, it does not tell you which ones since the gamma affects spacing between colors and make it non linear. This means that two 8-bit monitors show the same number of colors within a color-space yet they may not show all the same colors. It gets worse if you calibrate your graphics card rather than the display. The LUTs loaded into the graphics card map an 8-bit input from the OS to the input that is supported by the monitor (usually 8 or 10-bits), in the worse case of an 8-bit monitor, after calibration it will not even use all possible colors which is why this often results in banding! It is immensely better to calibrate the monitor which can have 12 or 14-bit Hardware LUTs, so they can map a full 8-bit input to the bit-depth of the display without loss of the number of possible colors shown.