An animal trainer wants to arrange 5 lions and 4 tigers in a single row so that no two lions stand next to each other. In how many distinct arrangements can this be done?

Introduction / Context:
This problem is a classic example of permutations with separation constraints, where certain objects (lions) are not allowed to be adjacent. You must use the idea of placing one type of animal first and then inserting the others into the available gaps to ensure the required separation condition is satisfied.

Given Data / Assumptions:

• There are 5 lions and 4 tigers.
• All lions are considered distinct, and all tigers are considered distinct.
• The animals are to be arranged in a row.
• No two lions are allowed to stand next to each other.

Concept / Approach:
A standard method for such problems is to first arrange the animals that have no restriction between themselves, and then place the restricted animals (the lions) into the gaps created. Here, if we first arrange the 4 tigers, they will create 5 potential gaps for the lions: before the first tiger, between each pair of tigers, and after the last tiger. To avoid two lions being adjacent, we must place exactly one lion in each gap because we have exactly 5 lions and 5 gaps.

Step-by-Step Solution:
Step 1: Arrange the 4 distinct tigers in a row. Number of ways = 4! = 24.Step 2: The 4 tigers create 5 gaps: _ T _ T _ T _ T _.Step 3: Place one lion in each gap to ensure no two lions are adjacent. Since we have 5 lions and 5 gaps, each gap receives exactly one lion.Step 4: The 5 lions are distinct and can be arranged among these 5 positions in 5! = 120 ways.Step 5: Total valid arrangements = 4! * 5! = 24 * 120 = 2880.

Verification / Alternative check:
If you tried to start with placing lions, you would have to ensure tigers separate them, which is less straightforward.The gaps method automatically guarantees that no two lions stand together.

Why Other Options Are Wrong:
2800 and 2600 are close in magnitude but do not equal 4! * 5! and likely come from arithmetic errors.3980 is larger and does not correspond to any logical counting pattern in this setup.

Common Pitfalls:
Allowing two lions to occupy the same gap, which violates the non adjacency rule.Treating lions or tigers as indistinguishable when the problem context typically assumes them to be distinct individuals.Forgetting one of the outer gaps when counting possible positions for lions.

Final Answer:
The number of valid arrangements is 2880.