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The problem asks to determine how many ways 3 tasks can be assigned to a committee consisting of 4 men and 6 women, with the condition that both men and women are given tasks.

### Step 1: General Overview
You have a total of 10 committee members, 4 of whom are men and 6 of whom are women. There are 3 tasks to be assigned, and the constraint is that both men and women must receive assignments.

### Step 2: Task Distribution
For both men and women to receive tasks, at least 1 man and 1 woman must be assigned a task. This means there are two possible ways to split the tasks between men and women:
1. **Case 1:** Assign 1 task to a man and 2 tasks to women.
2. **Case 2:** Assign 2 tasks to men and 1 task to a woman.

### Step 3: Counting the Assignments
We will use combinations to count the ways in which these tasks can be assigned.

#### Case 1: 1 man and 2 women get the tasks
- Choose 1 man from 4 men: \( \binom{4}{1} = 4 \)
- Choose 2 women from 6 women: \( \binom{6}{2} = \frac{6 \times 5}{2} = 15 \)
- The tasks can be assigned to these 3 people in \( 3! = 6 \) ways.

The total number of ways in this case is:
\[
4 \times 15 \times 6 = 360
\]

#### Case 2: 2 men and 1 woman get the tasks
- Choose 2 men from 4 men: \( \binom{4}{2} = \frac{4 \times 3}{2} = 6 \)
- Choose 1 woman from 6 women: \( \binom{6}{1} = 6 \)
- The tasks can be assigned to these 3 people in \( 3! = 6 \) ways.

The total number of ways in this case is:
\[
6 \times 6 \times 6 = 216
\]

### Step 4: Total Number of Ways
Adding both cases together, the total number of ways to assign the tasks such that both men and women get assignments is:
\[
360 + 216 = 576
\]

### Final Answer:
There are **576 ways** to assign the 3 tasks so that both men and women receive assignments.

Would you like further details or have any questions about this process?

Here are 5 related questions for further exploration:
1. What happens if there are no restrictions on the gender of the people who get the tasks?
2. How would the solution change if there were 5 men and 5 women?
3. What if 4 tasks needed to be assigned instead of 3?
4. How does the factorial \(3!\) affect the total number of assignments?
5. Can you derive the number of ways if the committee had 3 men and 7 women?

**Tip:** In combinatorics, when you're dealing with task assignments, remember to consider both the selection of participants and the arrangement of tasks carefully.

Rework problem 16 from section 2.2 of your text, involving the assignment of tasks to the men and women of a committee. Assume that you have a committee of 10 members, made up of 4 men and 6 women. In how many ways can the 3 tasks be assigned so that both men and women are given assignments?

Learn how to assign 3 tasks to a committee consisting of 4 men and 6 women, ensuring both genders receive assignments. This problem involves key combinatorics concepts, including combinations and permutations. Suitable for students in Grades 10-12.

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