A family consisting of a mother, father and their 8 children ( 4 boys and 4 girls) are to be seated at a round table in a party. How many ways can this be done if the mother and father sit together and the males and females alternate?

Given a family of 10 people—mother, father, 4 boys, and 4 girls—arranged around a round table with alternating genders and the mother and father seated together.

The table alternates between 5 male and 5 female positions. Fix the mother in a female seat to account for circular symmetry. The father must occupy an adjacent male seat to satisfy the togetherness condition. The arrangement of the remaining 4 girls and 4 boys in the vacant female and male seats is $4! \times 4! = 576$.

The final count is $576$.