minimum number of ways in which clients can be coded =

single colours + combination of 2 colours

You can eliminate the choices A and B.

Say If John takes 4 single colours than various combinations of colours would be

4C2 = 6

i.e total (single + combination , 4+6)= 10 , that doesnot statisfy the condition.

if single colours = 5

5C2 = 10

i.e total = 15 that statisy the condition.

therefore minimum number of colours required = 5