SOLUTION: How many ways are there to arrange 4 male students and 3 female students in 10 seats around a circle such that each gender sits together?

In this case, you have actually a block of 4 males   with their 4 chairs,

                                a block of 3 females with their 3 chairs,

                                and the block of remaining 10-4-3 = 3 vacant chairs.


And you arrange these 1 + 1 + 1 = 3 blocks circularly around the circular table.


It gives you   = 2 distinguishable circular permutations  of 3 blocks.



In addition, you have  4! = 24 permutations inside the block of 4 males
                  and  3! =  6 permutations inside the block of 3 females.


You do not permute vacant chairs inside the block of 3 vacant chairs,
since vacant chairs are indistinguishable.


All this gives you  2 * 24 * 6 = 288 distinguishable circular permutations.