Q9:

It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible ?

Step 1: Analyzing the Positional Constraints

We are required to arrange a total of $n = 9$ individuals (5 men and 4 women) in a single row. The constraint specifies that the 4 women must occupy the even-numbered positions in the row [Given condition].

The available positions in the row are: $P_1, P_2, P_3, P_4, P_5, P_6, P_7, P_8, P_9$.

The even positions within this set of nine are: $P_2, P_4, P_6, P_8$. There are exactly 4 even places available, which matches the number of women to be seated [Logical parity match].

Step 2: Calculating Arrangements for Women

There are 4 women and 4 designated even positions. The number of ways to arrange $n$ distinct objects in $n$ positions is given by the factorial $n!$ [Fundamental Principle of Counting].

Number of ways to seat the women $= 4!$

$4! = 4 \times 3 \times 2 \times 1 = 24$ ways.

Step 3: Calculating Arrangements for Men

After the women are seated in the even positions, there remain $9 - 4 = 5$ odd positions for the 5 men ($P_1, P_3, P_5, P_7, P_9$).

The number of ways to arrange 5 distinct men in 5 distinct positions is given by $5!$ [Permutation principle].

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ ways.

Step 4: Applying the Fundamental Principle of Multiplication

According to the Fundamental Counting Principle, if one event can occur in $m$ ways and a second independent event can occur in $n$ ways, then the total number of ways the two events can occur in sequence is $m \times n$ [Multiplication Axiom].

Total arrangements $= (\text{Ways to arrange women}) \times (\text{Ways to arrange men})$

Total arrangements $= 4! \times 5!$

Total arrangements $= 24 \times 120 = 2880$.

Visual Representation of Occupancy

Final Solution: The total number of possible arrangements is 2880.