Colour is reproduced mixing 3 channels of primary colour (Red, Green, Blue) and the colour depth tells how much information can be stored per colour channel. As each bit can hold a value of either 1 or 0, and combining say 8 bits for one channel allows to store up to 2^8=256 different values.
If there is room for 8 bits per channel, the total number of colours will then be a combination of the 3 channel meaning 256*256*256 = more than 16 million combinations. So a 8 bit colour depth yields more than 16 million combinations.
Notice that if the image is monochromatic and e.g. only uses the red channel, then the number of combinations are dramatically reduced as the other channels are not used for storing information.
Related reading
What is the difference between resolution and bit-depth in photography?
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