In how many ways can an animal trainer arrange 6 lions and 5 tigers in a row so that no two lions are together?

This problem can be solved using the concept of permutations and combinations.

Step 1: Arrange the tigers first. There are 5 tigers, so they can be arranged in 5! (factorial) ways.

Step 2: Now, consider the spaces between the tigers and at the ends of the row. There are 6 such spaces ( T_T_T_T_T ).

Step 3: The lions need to be placed in these spaces such that no two lions are together. There are 6 lions and 6 spaces, so the lions can be placed in these spaces in 6C6 ways (which is 1, because there's only one way to choose 6 spaces from 6).

Step 4: The lions can be arranged among themselves in 6! ways.

So, the total number of ways the animal trainer can arrange the lions and tigers is 5! * 6C6 * 6! = 120 * 1 * 720 = 86400 ways.