Question

7 boys and 5 girls are to be seated around a circular table such that no two girls sit together is?

• A. \(126(5!)^{2}\)
• B. \(720(5!)\)
• C. \(720(6!)\)
• D. 720

Hint:

When arranging people in a circle, account for rotational symmetry by fixing one person’s position.

Answer 1

The Correct Option is A

We need to arrange 7 boys and 5 girls around a circular table such that no two girls sit together. Let's solve this step by step:

Step 1: Arrange the Boys

Since it's a circular arrangement, we can fix the position of one boy and arrange the remaining boys around him. Thus, the number of ways to arrange 7 boys around a circular table is given by:

\(6!\)

This simplifies to:

\(6! = 720\)

Step 2: Arrange the Girls

The condition that no two girls sit together implies that each girl must sit between two boys. We can imagine this as placing the girls in the gaps between the boys. With 7 boys, there are 7 gaps available.

We need to select 5 out of these 7 gaps to place the girls, and the number of ways to select 5 gaps out of 7 is given by:

\(\binom{7}{5} = \binom{7}{2} = 21\)

Step 3: Permute the Girls in the Chosen Gaps

Once we choose the gaps, we need to arrange the 5 girls in these selected gaps. The number of ways to arrange 5 girls is given by:

\(5!\)

This simplifies to:

\(5! = 120\)

Step 4: Calculate the Total Arrangements

The total number of ways to arrange the 7 boys and 5 girls such that no two girls sit together is calculated by multiplying the number of arrangements of the boys, the number of ways to choose the gaps for the girls, and the number of arrangements of the girls:

\(6! \times \binom{7}{5} \times 5!\)

Substituting the values, we have:

\(720 \times 21 \times 120\)

Simplifying this expression gives:

\(= 1512000\)

This matches the form of the correct answer given by the option:

\(126(5!)^{2}\)

Thus, the correct answer is indeed \(126(5!)^{2}\).