A group of 9 friends, including Albert and Ben, are having dinner at Albert's house. They sit in two groups: a row of 4 on a couch and a group of 5 at a round dining table with 5 identical seats. Find the number of ways they can sit if

1. Find the number of ways to select and arrange 4 people on the couch.

Number of ways to choose 4 people from 9 and arrange them in a row on the couch = $^9P_4 = 3024$

2. Find the number of ways to arrange the remaining 5 people at the round table.

The remaining 5 people are seated at a round table. Since the seats are identical, this is a circular arrangement.
Number of ways to arrange 5 people in a circle = $(5-1)! = 4! = 24$

3. Calculate the total number of unrestricted seating arrangements.

Total number of ways = $3024 \times 24 = 72576$

1. Calculate the number of arrangements where Albert and Ben sit together on the couch.

Case 1: Albert and Ben are on the couch.

• Choose 2 other friends out of 7 to join them: $\binom{7}{2}$
• Arrange the 4 people on the couch, treating Albert and Ben as a single unit: $3! \times 2!$
• Arrange the remaining 5 people at the round table: $(5-1)! = 4!$
  $\text{Number of ways} = \binom{7}{2} \times (3! \times 2!) \times 4!$
  $= 21 \times 12 \times 24 = 6048$

2. Calculate the number of arrangements where Albert and Ben sit together at the round table.

Case 2: Albert and Ben are at the round table.

• Choose 3 other friends out of 7 to join them: $\binom{7}{3}$
• Arrange the 5 people at the round table, treating Albert and Ben as a single unit: $(4-1)! \times 2! = 3! \times 2!$
• Arrange the remaining 4 people on the couch: $4!$
  $\text{Number of ways} = \binom{7}{3} \times (3! \times 2!) \times 4!$
  $= 35 \times 12 \times 24 = 10080$

3. Sum the number of arrangements from both cases to find the total.

$\text{Total number of ways} = 6048 + 10080 = 16128$

1. Find the number of ways if Albert and Ben both sit on the couch but not beside each other.

Number of ways to choose 2 remaining friends out of 7 for the couch: $^7C_2 = 21$

Total ways to arrange 4 people on the couch: $4! = 24$
Ways Albert and Ben sit together on the couch (treating them as a single unit): $3! \times 2! = 12$
Ways they sit on the couch but not together: $24 - 12 = 12$

Ways to arrange the remaining 5 friends at the round table: $(5-1)! = 24$

Total ways for this case = $21 \times 12 \times 24 = 6048$

2. Find the number of ways if Albert and Ben both sit at the round table but not beside each other.

Number of ways to choose 4 friends out of 7 for the couch (leaving 3 for the table): $^7C_4 = 35$

Ways to arrange the 4 friends on the couch: $4! = 24$

Total ways to arrange 5 people at the round table: $(5-1)! = 24$
Ways Albert and Ben sit together at the round table (treating them as a single unit among 4 total units in a circle): $(4-1)! \times 2! = 3! \times 2 = 12$
Ways they sit at the round table but not together: $24 - 12 = 12$

Total ways for this case = $35 \times 24 \times 12 = 10080$

3. Calculate the total number of arrangements by summing both mutually exclusive cases.

Total ways = $6048 + 10080 = 16128$