The answer is basically infinity, But the number of colours we see is a different story.
Psycho Physicists; who study human responses believe they have figured out the total number of colours the human eye can distinguish.
A study done by these Psycho Physicists, says that the eye can see one thousand shades of light, as well as one hundred levels of red-green and one hundred levels of yellow-blue.
This means its 1000 x 100 x 100 = 10,000,000. (ten million)
Ten million shades of color!
However, they all stem from 11 main colors: red, orange, yellow, green, blue, purple, pink, brown, gray, black and white.
The rest are just combinations of these colors.
It may also interest you to know that today’s standard computer screen shows over 16 million hues of color for a single full color image.