Each picture element has a number of bits which are 'mapped' to it. The number of bits dedicated (or 'mapped') to each pixel determines the number of colours which the pixel can be assigned. For example: two bit colour allows for four unique colours; while eight bit colour gives us 256 different colour values.
The number of colours available in an image - its "colour depth" - can therefore be expressed in terms of the number of bits mapped to each pixel. If the number of bits in an image or display is n then it can have up to 2n colours. So four bit colour gives us 24 colours: 16. Eight bit colour gives 28, that is, 256 colours. And so on.
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