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The question asks to form a line with 4 men and 3 women, and two correct answers must be chosen based on the given alternatives.

### Analysis of the Alternatives:

1. **Em nenhuma posição temos \(4! \times 3!\) possibilidades.**
   - This refers to forming two separate groups of men and women and permuting them separately. There are \(4!\) ways to arrange 4 men and \(3!\) ways to arrange 3 women. However, the phrasing implies that they don't mix, so this doesn't apply to a scenario where both genders alternate or mix freely.

2. **Intercalando homens e mulheres temos \(4 \times 1 \times 3 \times 1 \times 2 \times 1 \times 1 = 24\) possibilidades.**
   - This refers to alternating men and women in the line. The first position is occupied by a man, the second by a woman, and so on. This reasoning holds: 4 options for the first man, 1 for the woman, 3 for the second man, and so forth. The correct calculation results in 24 possibilities. **This is correct.**

3. **Em qualquer posição temos \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\) possibilidades.**
   - This alternative calculates the number of possible arrangements when there are no restrictions at all. Since there are 7 people in total, the total number of permutations is indeed \(7!\), which equals 5040. **This seems correct** if no restrictions are applied.

4. **Intercalando homens e mulheres temos \(4 \times 3 \times 3 \times 2 \times 2 \times 1 \times 1 = 144\) maneiras diferentes.**
   - This represents a different way of calculating the alternating arrangement, but it's not correct. The given logic implies different steps than what would occur in alternating seating. The correct answer for alternating genders is 24, not 144.

5. **Em qualquer posição temos \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\) possibilidades.**
   - This mistakenly considers 6 people instead of 7, which is incorrect since there are 7 individuals.

### Conclusion:
The two correct answers are:
- **Intercalando homens e mulheres temos \(24\) possibilidades.**
- **Em qualquer posição temos \(7! = 5040\) possibilidades.**

Would you like further details or explanations?

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Here are 5 relative questions for you:
1. How does the number of permutations change when additional people are added to the group?
2. What happens to the total number of possibilities if we include restrictions such as specific people not being able to stand next to each other?
3. Can the factorial concept be applied to other fields outside of mathematics? If so, how?
4. How can combinatorics help in organizing groups for real-world applications like events or seating arrangements?
5. What is the difference between permutations and combinations, and how does each apply to different types of problems?

**Tip:** When working with permutations, always check if there are restrictions or conditions that reduce the total possibilities.

Deseja-se formar uma fila com 4 homens e 3 mulheres. Observando a disposição da fila descrita nas alternativas, assinale as duas corretas.

Solve the problem of arranging 4 men and 3 women in a line with various conditions. This problem covers combinatorics concepts like permutations, alternating arrangements, and factorial calculations. Suitable for Grades 10-12 students.

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