John has 12 clients and he wants to use color coding to identify each client. If either a single color or a pair of two different colors can represent a client code, what is the minimum number of colors needed for the coding? Assume that changing the color order within a pair does not produce different codes.
I think 4 colors would be sufficient
With 2 colors, he can code 2P1 + 2P2 = 4 clients. If a,b are colors then a, b, ab, ba.
With 3 colors 3P1 + 3P2 = 9 clients
With 4 colors 4P1 + 4P2 = 16 clients.
So 4 should be the answer.
If order of color does not matter (which is nonsensical for color code) then 5 colors needed
with 2 colors 2C1 + 2C2 = 2
with 3 colors 3C1 + 3C2 = 6
with 4 colors 4C1 + 4C2 = 10
with 5 colors 5C1 + 5C2 = 15