What are the number of ways of seating $n$ people around a circular table labelled from $1$ to n such that $3$ of them must be seated together?

The full question (How to count by Robert A. Beeler) is as follows:

Suppose that we have $n$ people (where $n \geq 3$) to be seated around a circular dinner table with $n$ seats labelled $1, \ldots, n$. Three of the guests, Alice, Bob, and Chad must all be seated together in some order. In how many ways can this be done?

I have found a solution but that doesn't match with the answer and unfortunately, I cannot find a problem in my logic, here it is:

i. We can group Alice, Bob and Chad as a single object O, then the total number of objects we have to arrange is $$(n - 3 + 1) = (n-2)$$

[$-3$ from removing the $3$ people and $+1$ from adding that grouped object]

ii. Because the table is labelled, it means that every single arrangement will be unique, thus there won't be any symmetries in reflections or rotations.

ii. The total number of ways comes out to be $$(n-2)!$$

iii. Now these $3$ people can be arranged themselves in the group as $3! = 6$

Thus my solution leads to the number of ways as $$6(n-2)!$$

Here is the book's solution:

i. Choose three places for Alice, Bob, and Chad. Since the table is circular, there are $n$ ways to do this.

ii. Seat Alice, Bob, and Chad. There are $3! = 6$ ways to do this.

iii. Seat the remaining guests. There are $(n − 3)!$ ways to do this. By the Multiplication Principle, there are $$6n(n − 3)!$$

I do understand this solution, but I don't understand why my solution doesn't match with the book's solution.

In your attempt, when you opted to treat the three guests who must sit together as a block, you reduced the number of objects from $n$ to $n - 2$. Consequently, you had $n - 2$ ways to place the block. However, there are actually $n$ possible starting positions for the block. To compensate for this, you must multiply your answer by $\frac{n}{n - 2}$, which yields $$\frac{n}{n - 2} \cdot 6(n - 2)! = 6n(n - 3)!$$ which agrees with the answer in you book.