In how many different arrangements can 7 white suited persons and 5 black suited persons sit in a round table of 12 seats so that none of the black suited persons gets to sit next to each other?
The way I tried to approach this is by considering the possible arrangements of seating the black suited people and the white suited people as two groups.
And for each of these arrangements the white suited guys can rearrange their positions in 7! ways and for each of those rearrangements the black suited guys can rearrange their positions in 5! ways. So, the number of total possible seating arrangements we get are $3\times7!\times5!$
Is this approach correct? Are there other better ways to approach the problem?