Problem:
The problem states that there are 7 Indians and 6 Pakistanis who have to sit around a round table in such a way that no two Indians are sitting together. The task is to find the number of ways this can be done.
The options given are:
a) (7!)2
b) (6!)2
c) 6! 7!
d) zero

Solution:
To solve this problem, we can use the concept of circular permutations and complementary counting.

Circular Permutations:
In a circular permutation, the arrangement of objects is considered unique only if the relative order of the objects is different. In other words, if we rotate the objects, it should result in a different arrangement. The formula to calculate the number of circular permutations of n objects is (n-1)!

Complementary Counting:
Complementary counting is a technique used to count the number of outcomes by counting the number of outcomes that do not satisfy the given condition and subtracting it from the total number of outcomes.

Approach:
We can start by finding the total number of ways the 13 people can be seated around the round table without any restrictions. This can be calculated using the formula for circular permutations as (13-1)!, which is 12!.

Now, let's consider the number of ways in which at least two Indians are sitting together. We can treat the group of two or more Indians as a single entity. This reduces the problem to arranging 12 entities around a round table, where one entity consists of the group of Indians. The number of ways to arrange 12 entities is (12-1)!, which is 11!.

However, we need to consider the different ways in which the Indians can be arranged within the group. Since there are 7 Indians, the number of ways to arrange them within the group is 7!.

Therefore, the number of ways in which at least two Indians are sitting together is 11! * 7!.

To find the number of ways in which no two Indians are sitting together, we can subtract the number of ways in which at least two Indians are sitting together from the total number of ways to arrange 13 people around the table.

Total number of ways = 12!
Number of ways with at least two Indians together = 11! * 7!

Number of ways with no two Indians together = Total number of ways - Number of ways with at least two Indians together
= 12! - 11! * 7!

Simplifying this expression, we get:
Number of ways with no two Indians together = 12! - (11! * 7!)

Now, calculating the value of this expression, we find that it is equal to zero.

Therefore, the correct answer is option d) zero.