Circular Permutations

Circular permutations are arrangements where objects are placed in a circle rather than a straight line. In circular permutations, the number of arrangements is generally given by (n-1)!, where n is the number of objects being arranged.

In this problem, we have 5 Americans and 5 Indians who need to be seated in alternative positions around a circular table. Let's break down the solution into steps.

Step 1: Fix one person's position

Since it is a circular table, we can fix the position of one person, say an American. We have 5 Americans to choose from, so we have 5 options for this step.

Step 2: Arrange the remaining Americans

We have 4 Americans left to arrange, and they need to be seated in alternate positions with the Indians. This means that we have 5 seats available for the Americans (since the Indian seats will alternate with them), and we need to arrange the 4 Americans in these seats.

The number of ways to do this is (4-1)! = 3!, since we have 4 objects to arrange in a straight line, but we have to divide by 2 to account for the fact that the Americans can be arranged in opposite orders (e.g. ABCD or DCBA).

Step 3: Arrange the Indians

We have 5 Indians left to arrange, and they need to be seated in alternate positions with the Americans. This means that we have 5 seats available for the Indians (since the American seats will alternate with them), and we need to arrange the 5 Indians in these seats.

The number of ways to do this is (5-1)! = 4!, since we have 5 objects to arrange in a straight line, but we have to divide by 2 to account for the fact that the Indians can be arranged in opposite orders (e.g. EFGHI or IHGFE).

Step 4: Multiply the possibilities

Now we have to multiply the possibilities from Steps 1, 2, and 3 to get the total number of arrangements. The number of ways to seat the Americans and Indians in alternate positions around the circular table is:

5 x 3! x 4! = 5 x 6 x 24 = 720

Therefore, the correct answer is option C, 4! x 5!.