There are 7 colours in the rainbow; red, orange, yellow, green, blue, indigo and violet.

The traditional answer is seven distinct colours. However, I doubt that a plurality of people would identify them this way without relying on memory. For example, how often does anyone answer "indigo" when asked what colour an item is? It's not really part of the lexicon of the average person outside of art and science. It might be more accurate to define the rainbow as red, orange, yellow, green, blue, and violet. Or, if you include indigo, might as well also include cyan and lime as well, since those are less obscure. But, to be completely thorough, the most correct answer is that there is an uncountable number of colours in the rainbow. A rainbow is represented by all components of broad-spectrum white light. This is generally said to be a continuum of spectral colours from about 380 nm to 700 nm. So, 400 nm is one colour, and, 401 nm is technically another colour. How many numbers are there between 380 and 700? Well, it's infinity, but it's a particularly dense sort of infinity, since we can choose a starting point, but cannot even count the second number. In other words, what number comes directly after 380? Is it 381? No, because decimal values still count. 380.1? 380.01? 380.001? etc...

Richard Of York Gave Battle In Vain. A mnemonic I'll never forget.