Five girls and five boys are to be seated on a bench with the boys and girls alternating. Find the number of ways of their seating. In how many different ways could they sit around a circular table so that the boys and girls alternate ?

Answer

To solve the problem of seating five girls and five boys on a bench with alternating genders, we can break it down into two parts:

### Part 1: Seating on a Bench

1. **Arrange the Boys**:
- We have 5 boys who need to be seated. The number of ways to arrange 5 boys is given by the factorial of the number of boys.
\[ \text{Ways to arrange boys} = 5! = 120 \]

2. **Arrange the Girls**:
- After seating the boys, we need to seat the girls in the remaining positions. Since the seating must alternate, there will also be 5 positions for the girls. The number of ways to arrange 5 girls is also given by the factorial of the number of girls.
\[ \text{Ways to arrange girls} = 5! = 120 \]

3. **Total Arrangements**:
- Since the arrangements of boys and girls are independent, we multiply the number of arrangements of boys by the number of arrangements of girls.
\[ \text{Total arrangements} = 5! \times 5! = 120 \times 120 = 14400 \]

### Part 2: Seating Around a Circular Table

1. **Fix One Boy**:
- When arranging people around a circular table, we can fix one person to eliminate the effect of rotations. Here, we can fix one boy in one position. This leaves us with 4 remaining boys to arrange.
\[ \text{Ways to arrange remaining boys} = 4! = 24 \]

2. **Arrange the Girls**:
- After seating the boys, there will be 5 gaps between the boys where the girls can sit. We can arrange the 5 girls in these gaps.
\[ \text{Ways to arrange girls} = 5! = 120 \]

3. **Total Arrangements**:
- The total arrangements around the circular table will be the product of the arrangements of the remaining boys and the arrangements of the girls.
\[ \text{Total arrangements} = 4! \times 5! = 24 \times 120 = 2880 \]

### Final Answers:
- The number of ways to seat the boys and girls on a bench is **14400**.
- The number of ways to seat them around a circular table is **2880**.