Step-by-step Solution:

Step 1: Understanding the Given Problem
We have 4 Indians, 3 Americans, and 2 Britishers, making a total of: \[ 4 + 3 + 2 = 9 \] These 9 people are to be arranged around a round table such that the two Britishers never sit together.
Step 2: Arranging the Remaining People
First, we arrange the 7 non-British people (4 Indians + 3 Americans) around a circular table. Since circular permutations are considered, the number of ways to arrange these 7 people is: \[ (7-1)! = 6! \] Step 3: Placing the Britishers
Once the 7 non-British people are arranged, they create 7 spaces (gaps) where the 2 Britishers can sit.
To ensure they do not sit together, we choose 2 out of these 7 spaces for the Britishers. This is a permutation problem, as the order in which the Britishers sit matters: \[ {}^7P_2 = \frac{7!}{(7-2)!} = \frac{7!}{5!} = 7 \times 6 = 42 \] Step 4: Final Calculation
Thus, the total number of ways to arrange the people such that the Britishers do not sit together is: \[ 6! \times {}^7P_2 \]