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

Answer 1

Answer: (A)

$126(5!)^{2}$

Answer 2

The correct answer is (A) :
First, we need to find the total number of ways to seat all 12 people around the circular table, which is (12-1)! = 11! since we can fix one person's position as a reference.
Next, we need to subtract the number of ways that two or more girls sit together. We can approach this by treating the five girls as a block and permuting them first, which can be done in 5! ways.
Then we can insert this block of girls in the 8 spaces between the 7 boys or at the beginning or end of the line of boys, which gives us 9 positions to place the block of girls. Once the block of girls is placed, we can permute the 7 boys in 7! ways. Therefore, the total number of ways that two or more girls sit together is 5! × 9 × 7!
$\therefore$ the number of ways that no two girls sit together is 11! - 5! × 9 × 7! = 126(5!)2.