To solve this, we need to understand the implications of seating people around a circular table and the presence of 11 identical chairs.

Key Observations:

1. Circular Seating: In circular seating, the arrangement is considered rotationally equivalent. That is, rotating a given arrangement does not create a new arrangement.
2. Extra Chair: There are 11 identical chairs but only 10 people to be seated. This means one chair will remain empty, but since the chairs are identical, the position of the empty chair doesn't create a new unique arrangement (the arrangement is invariant under rotation).

Step-by-Step Solution:

Step 1: Fix One Person

For circular seating problems, we typically fix one person to break the rotational symmetry. After fixing one person in a specific chair, the remaining $9$ people can be seated in the other $9$ chairs.

Step 2: Seat the Remaining People

Once the first person is fixed, the remaining $9$ people can be seated in the remaining $9$ chairs (excluding one chair that remains empty). The number of ways to arrange $9$ people in $9$ chairs is $9!$.

Step 3: Conclusion

Thus, the total number of ways to seat the $10$ people across $11$ identical chairs (with one person fixed) is simply:

$9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880$

Final Answer:

There are 362,880 ways to seat 10 people across 11 identical chairs placed equally around a circular table.

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Do you want more details or have any questions?

Here are 5 related questions you might explore:

1. How would the answer change if the chairs were not identical?
2. What if there were only 10 chairs for 10 people at a circular table?
3. What if two of the 10 people must sit next to each other?
4. How would the solution change if it were a straight line of chairs instead of a circular arrangement?
5. How many ways can 10 people be seated if 3 specific people must be seated together?

Tip: For circular permutations, remember that fixing one element breaks the rotational symmetry, simplifying the counting process.