Question

question 8 (3 points) how many ways can 9 people be seated at a round table, if marko refuses to sit next to juan?

Explanation:

Step1: Find total circular permutations

For \( n \) people in a round table, total permutations are \( (n - 1)! \). Here \( n = 9 \), so total ways \( = (9 - 1)! = 8! = 40320 \).

Step2: Find permutations where Marko and Juan sit together

Treat Marko and Juan as a single entity. Now we have \( 8 \) entities (the pair + 7 others) in a round table. Permutations of these \( 8 \) entities: \( (8 - 1)! = 7! \). Marko and Juan can be arranged within the pair in \( 2! \) ways. So total ways with them together: \( 7! \times 2! = 5040 \times 2 = 10080 \).

Step3: Subtract unwanted from total

Ways where Marko does not sit next to Juan: Total - Together \( = 8! - 7! \times 2! = 40320 - 10080 = 30240 \).

Answer:

\( 30240 \)