In the figure below, 4 of the 6 disks are to be colored black and 2 are to be colored white. Two colorings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. There are only four such colorings for the given two colors, as shown in Figure 1. In how many ways can we color the 6 disks such that 2 are colored black, 2 are colored white, 2 are colored blue wit

Step-by-step solution

We need to change the colour of any two black disks to blue. From given four coloring patterns, in (i), (iii) & (iv) we can select any two black disks in ^4 C₂ ways, but only two selections will be unique and other four combinations will be reduced into two distinct combinations (either due to rotation or reflection). In (ii) we can select any two black disks in ^4 C₂ ways. So in each given coloring there exist only four possibilities, hence number of required coloring patterns =4 3+6=18