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.