Seating Arrangements for Grouped Preferences on a Long Table

Imagine a long table with 12 seats, 6 on each side. You have 12 people to seat, with 4 of them preferring one side of the table and 3 preferring the other. The challenge is to determine the number of seating arrangements that accommodate everyone's preferences. Let's break down the process and calculate the total number of possible arrangements.

Assigning People to Sides

We have 12 people in total, with 4 who must sit on one side (let's call this Side A) and 3 who must sit on the other side (Side B). This leaves us with 5 people (Side C) who can sit on either side.

Seating Arrangements

Side A

Side A has 6 seats, with 4 already occupied by the 4 people who want to sit there. We need to choose 2 out of the 5 remaining people to fill the remaining seats on Side A.

The number of ways to choose 2 people from 5 is given by the binomial coefficient (binom{5}{2}).

[binom{5}{2} frac{5!}{2!(5-2)!} frac{5 times 4}{2 times 1} 10]

After choosing 2 people for Side A, we have 6 people seated there.

Side B

Side B also has 6 seats, with 3 already occupied by the 3 people who want to sit there. The remaining 3 seats will be filled by the 3 people who were not chosen for Side A (the remaining people from the original group of 5).

Arranging the Seating

Once we have assigned the people to each side, we need to arrange them. The number of arrangements for Side A with 6 people is (6!) and the number of arrangements for Side B with 6 people is also (6!).

[boxed{6! 720}]

Calculating Total Arrangements

The total number of seating arrangements can be calculated by multiplying the number of ways to choose people for Side A by the arrangements for both sides:

[text{Total arrangements} binom{5}{2} times 6! times 6! 10 times 720 times 720]

[text{Total arrangements} 10 times 518400 5184000]

Therefore, the total number of seating arrangements is 5,184,000.

Final Answer: 5,184,000 arrangements