Step-by-step Solution:

We are given 4 Indians, 3 Americans, and 2 Britishers, making a total of: \[ 4 + 3 + 2 = 9 \] people to be arranged around a round table. However, we need to find the number of ways to arrange them such that the three Americans always sit together. Step 1: Treating the Three Americans as One Block Since the three Americans must sit together, we treat them as a single unit or a block. This reduces the number of units to be arranged in a circle: \[ (4 + 2 + 1) = 7 \] Thus, the number of circular permutations of these 7 units is: \[ (7-1)! = 6! \] Step 2: Arranging the Three Americans Within Their Block Within the block, the 3 Americans can be arranged among themselves in: \[ 3! \] ways. Final Calculation Thus, the total number of valid arrangements is: \[ 6! \times 3! \] Final Answer: \[ \boxed{6! \times 3!} \]