There are 39500 ways for the 8 men to stand in line such that no men from Tribe A are next to each other.
How to find amount?
Consider the restriction that no men from Tribe A should be next to each other. Use the principle of permutations.
First, consider the total number of ways the 8 men can stand in line without any restrictions, then the ways Tribe A men can be together. Since there are 3 men from Tribe A, they can be arranged among themselves in 3! ways.
The men from Tribe B and C can also be arranged among themselves in 5! ways.
Now, the total number of arrangements where men from Tribe A are together is:
3! × 5!.
To find the total number of arrangements where no men from Tribe A are next to each other, subtract the above result from the total number of arrangements without any restrictions:
Total arrangements without restriction - Arrangements with Tribe A together = 8! - (3! × 5!)
8! - (3! × 5!) = 40320 - (6 × 120)
= 40320 - 720
= 39500
Therefore, there are 39500 ways for the 8 men to stand in line such that no men from Tribe A are next to each other.