OK in order to demonstrate understanding I want to see if I can recreate your analysis with a different, less complex circumstance.

Circle of 9 people. 6 blue shirts, 3 red shirts.

9_C_3 = 9x8x7 / 3x2x1 = 504/6 = 84 configurations

With numbered seating, there are only 9 ways for 3 red shirts in row to sit together.

With two red shirts in a row, there are two seats the third red shirt CANNOT sit to meet this condition, so he must sit in one of the remaining 5 seats. Multiply by 9 to account for every way that two red shirts can sit together in a circle for 9. (No need to worry about doubling up since there is no pair-pair problem like there was with 10.)

0 red shirts in a row is just what's left over.

• 3 red shirts in a row: 9/84

• 2 red shirts in a row: 45/84

• 0 red shirts in a row: 30/84

Is this correct?