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This problem asks about selecting a committee of 4 people from a group of 7 women and 5 men, with different conditions on how the committee is to be formed.

Let's break it down into parts:

1. **Part 1: Total number of ways to select a committee of 4 people.**
   - You have 7 women and 5 men, so the total number of people is \( 7 + 5 = 12 \).
   - The number of ways to choose 4 people out of 12 is given by the combination formula:
   \[
   \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495.
   \]
   So, there are 495 ways to select a committee of 4 people.

2. **Part 2: Number of ways to select a committee with 2 men and 2 women.**
   - The number of ways to choose 2 women from 7 is:
   \[
   \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21.
   \]
   - The number of ways to choose 2 men from 5 is:
   \[
   \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10.
   \]
   - Therefore, the total number of ways to select 2 men and 2 women is:
   \[
   \binom{7}{2} \times \binom{5}{2} = 21 \times 10 = 210.
   \]

3. **Final Answer:**
   - The total number of ways to select a committee of 4 people is **495**.
   - The number of ways to select a committee with exactly 2 men and 2 women is **210**.

Would you like more details or explanations?

Here are five related questions:
1. How many ways can a committee of 3 people be selected from the same group?
2. How many ways can a committee of 5 people be selected if there must be at least 3 women?
3. What if there are no restrictions on the gender of the committee members? How would the problem change?
4. How would the result change if there were only 4 men and 6 women in the group?
5. How many ways can the committee be formed if there must be exactly 3 women?

**Tip:** When working with combinations, remember the formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items and \( r \) is the number of items chosen.

There are 7 women and 5 men in a department. How many ways can a committee of 4 people be selected? How many ways can this committee be selected if there must be 2 men and 2 women on the committee? How many ways can this committee be selected if there must be at least 2 women on the committee?

Explore how to select a committee of 4 people from 7 women and 5 men using combinatorics. Learn how to calculate different scenarios, including when 2 men and 2 women must be selected, or when there must be at least 2 women on the committee. Suitable for high school students.

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