Step-by-step Solution:

We are given a total of 4 Indians, 3 Americans, and 2 Britishers, which means there are: \[ 4 + 3 + 2 = 9 \] people to be arranged around a round table. Step 1: Formula for Circular Permutations For \( n \) distinct objects arranged in a circle, the number of ways to arrange them is: \[ (n-1)! \] This is because in circular permutations, one object is fixed to eliminate equivalent rotations. Step 2: Applying the Formula Since we have 9 people, the number of ways to arrange them in a circular manner is: \[ (9-1)! = 8! \] Final Answer: \[ \boxed{8!} \]