Five men and five women are to be arranged in a row while seating in a party.
Quantity I: Number of ways of arranging 5 men and 5 women such that on two men or women are adjacent to each other.
Quantity II: Number of ways of arranging 5 men and 5 woment such that all men sit together.

Options

A. Quantity I `gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge` Quantity II
D. Quantity I `le` Quantity II

Explanation

To solve the problem, we need to calculate the two quantities separately.

### Quantity I: Number of ways of arranging 5 men and 5 women such that no two men or women are adjacent to each other.

1. **Arrange the Women**:
First, we can arrange the 5 women. The number of ways to arrange 5 women is given by \(5!\).
\[ 5! = 120 \]

2. **Identify Gaps for Men**:
Once the women are arranged, they create gaps where men can be placed. For 5 women, there are 6 gaps (one before each woman, one after the last woman).
- Example: _ W _ W _ W _ W _ W _
Here, underscores represent the gaps.

3. **Choose Gaps for Men**:
We need to select 5 out of these 6 gaps to place the men. The number of ways to choose 5 gaps from 6 is given by \( \binom{6}{5} \).
\[ \binom{6}{5} = 6 \]

4. **Arrange the Men**:
The 5 men can be arranged in the selected gaps in \(5!\) ways.
\[ 5! = 120 \]

5. **Total Arrangements for Quantity I**:
Now, we multiply the number of arrangements of women, the number of ways to choose gaps, and the arrangements of men.
\[ \text{Total for Quantity I} = 5! \times \binom{6}{5} \times 5! = 120 \times 6 \times 120 = 86400 \]

### Quantity II: Number of ways of arranging 5 men and 5 women such that all men sit together.

1. **Treat Men as a Single Unit**:
Since all men must sit together, we can treat the group of 5 men as one single unit or block. Therefore, we have 6 units to arrange: the block of men and the 5 individual women.

2. **Arrange the Units**:
The number of ways to arrange these 6 units (1 block of men + 5 women) is \(6!\).
\[ 6! = 720 \]

3. **Arrange the Men Within the Block**:
The 5 men within the block can be arranged among themselves in \(5!\) ways.
\[ 5! = 120 \]

4. **Total Arrangements for Quantity II**:
Now, we multiply the arrangements of the units by the arrangements of the men within the block.
\[ \text{Total for Quantity II} = 6! \times 5! = 720 \times 120 = 86400 \]

### Conclusion:
- Quantity I = 86400
- Quantity II = 86400

Thus, both quantities are equal.