In how many ways can 5 boys and 5 girls be seated at a round table so that no two girls may be together ?

Answer

To solve the problem of seating 5 boys and 5 girls at a round table such that no two girls are seated together, we can follow these steps:

### Step 1: Fix the Position of Boys
Since we are dealing with a round table, we can fix one boy's position to eliminate the effect of rotations. This means we will arrange the remaining 4 boys around the table.

**Calculation:**
- The number of ways to arrange 4 boys is given by \(4!\).

### Step 2: Determine the Positions for Girls
Once the boys are seated, they create 5 gaps (or positions) where the girls can be placed. The arrangement will look like this: Boy - Gap - Boy - Gap - Boy - Gap - Boy - Gap - Boy - Gap - Boy.

Since we want to ensure that no two girls sit together, we can only place one girl in each gap.

### Step 3: Arrange the Girls in the Gaps
Now, we need to arrange the 5 girls in the 5 gaps created by the boys.

**Calculation:**
- The number of ways to arrange 5 girls is given by \(5!\).

### Step 4: Combine the Arrangements
To find the total number of arrangements, we multiply the number of ways to arrange the boys by the number of ways to arrange the girls.

**Final Calculation:**
\[ \text{Total arrangements} = 4! \times 5! \]

### Step 5: Calculate the Factorials
Now we can calculate the values of \(4!\) and \(5!\):
- \(4! = 24\)
- \(5! = 120\)

### Step 6: Calculate the Total
Now, we can compute the total number of arrangements:
\[ \text{Total arrangements} = 24 \times 120 = 2880 \]

Thus, the total number of ways to seat 5 boys and 5 girls at a round table such that no two girls are together is **2880**.

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