Number of ways 10 men and 4 women can sit at a round table so...
Number of ways 10 men and 4 women can sit at a round table so there is no block of 3 consecutive women
In how many ways can 10 men and 4 women sit at a round table so that there is no block of 3 consecutive women?
Since there is no restriction on men, I thought they should sit first. Since the table is round, I fixed one seat for 1 in 10 men and placed the rest of them 9! ways and then we have 10 gaps in which we can place 4 women, at most 2 in each. I think these are the possibilities:
1. One woman in each gap: we can choose those gaps in ways and then place women in 4! different ways.
2. 2 women in 1 gap and the remaining 2 in each of their own. We need 3 gaps altogether, which can be chosen in ways. Since women are different entities, I believe we should be able to place them in, again 4! ways.
3. 2 women in 2 gaps: 2 in each. We can choose 2 gaps in ways and again, if I'm not wrong, place them in 4! ways.
Therefore, my answer to the number of possible ways is: