Around a table are seated $m$ men and $f$ women. In how many ways can this be achieved if no two men sit together?
My son was given a variant of this (m=3 and f=8) on a test in highschool. The particular case is easy to count by hand but I could not find a general formula. Necklace counting results do not seem to apply directly, and I am trying to find a result without resorting to Pólya's enumeration theorem.
There are $(f-1)!$ ways in which women can seat around table. Now there are $f$ positions between women for men. So for first man there are $f$ positions to seat around. Now for second man there are $f-2$ positions to seat around. Simillarly for $m^{th}$ man there are remaining $f-m$ positions are available.
Hence by multiplication rule, total number of ways to sit around table are $(f-1)!*f*(f-1)*....*(f-m)$.