### Instruction: A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions?  (Two seatings are considered the same if one is a rotation of the other.).### Response: Since the table is circular, there is no fixed starting point or direction for the seating arrangement.
To avoid counting rotations as different seatings, I can fix one person's position and then arrange the others around them.
For example, I can imagine that the first Democrat always sits at the top of the table, and then count how many ways the other 9 politicians can be seated clockwise from them.
Once I fix the first Democrat's position, I have 9 choices for who sits to their right, then 8 choices for who sits next, and so on, until I have 1 choice for who sits to their left.
Multiplying these choices together gives me 9! ways to arrange the other 9 politicians.
This is the same as the number of permutations of 9 objects, which makes sense since I am permuting them around the circle.
Therefore, the answer is 9!.