Five men and four women are to be seated in a row such that all the four women occupy only the even numbered positions in the row. How many such linear arrangements are possible?

Introduction / Context:
This arrangement question combines permutations with a positional restriction. We have five men and four women to be seated in a row, with the condition that every woman must occupy an even position in the row.

Given Data / Assumptions:

• Total people = 5 men + 4 women = 9.
• Positions in the row are numbered 1 to 9 from left to right.
• Women must sit only in even positions (2, 4, 6 and 8).
• All men and women are distinct individuals.

Concept / Approach:
We treat seating as a two stage process. First, we seat the four women in the four available even positions. Then we seat the five men in the remaining five positions (all of which are odd positions). Since all individuals are distinct, we use factorial counts for each group and multiply the results.

Step-by-Step Solution:
Step 1: Identify the even positions in the row: 2, 4, 6 and 8. There are 4 such positions. Step 2: The four women must occupy these four even positions. Step 3: Number of ways to arrange 4 distinct women in 4 distinct positions = 4! = 24. Step 4: The remaining positions are 1, 3, 5, 7 and 9, which are 5 positions to be filled by the 5 men. Step 5: Number of ways to arrange 5 distinct men in 5 positions = 5! = 120. Step 6: Since choices are independent, total arrangements = 4! * 5! = 24 * 120 = 2880.

Verification / Alternative check:
You could reverse the order by first placing men in the 5 odd positions and then women in the even positions, but the final multiplication remains 5! * 4! which is 2880. This confirms the earlier calculation and shows that the order of the two stages does not affect the final count.

Why Other Options Are Wrong:
1440 and 720 correspond to using either 4! alone or 5! alone or dividing incorrectly. 2020 does not correspond to any natural factorial combination here and reflects a miscalculation.

Common Pitfalls:
Errors often arise from treating the problem as if there were fewer restrictions or from assuming that some positions are indistinguishable. Another common issue is accidentally swapping counts for men and women or missing one group in the multiplication.

Final Answer:
The number of valid linear arrangements is 2880.