how many colors
Used to ask about the number of colors, not to ask 'what color' (which is 何色 read なにいろ).
How many colors are used in this picture?
How many colors does a rainbow have?
Read なにいろ, means 'what color'. 何色 read なんしょく asks about the number of colors.
multicolored printing
Compound of 何 (なん, 'how many') + 色 (しょく, 'color').