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The problem asks how many ways three research departments, with 9, 10, and 6 members, respectively, can each select a delegate and an alternate to represent them at a conference.

Each department needs to select two members, a delegate and an alternate. The number of ways to select two people from a group is calculated by first selecting a delegate (any of the members) and then selecting an alternate (from the remaining members).

1. **Department 1 (9 members)**:
   - Ways to select a delegate: 9 choices
   - Ways to select an alternate: 8 remaining choices
   - Total for Department 1: \( 9 \times 8 = 72 \)

2. **Department 2 (10 members)**:
   - Ways to select a delegate: 10 choices
   - Ways to select an alternate: 9 remaining choices
   - Total for Department 2: \( 10 \times 9 = 90 \)

3. **Department 3 (6 members)**:
   - Ways to select a delegate: 6 choices
   - Ways to select an alternate: 5 remaining choices
   - Total for Department 3: \( 6 \times 5 = 30 \)

Finally, multiply the results for each department to find the total number of ways the selections can be made:

\[
72 \times 90 \times 30 = 194,400
\]

Thus, the total number of ways this can be done is **194,400**.

Would you like a deeper explanation of any part of the solution? Here are some related questions:

1. What if two members from the same department could not be selected?
2. How would the answer change if each department had the same number of members?
3. How would the solution differ if there were more departments?
4. What if a department could select the same person as both delegate and alternate?
5. How would the calculation change if only one representative were needed?

**Tip:** When counting arrangements, always consider whether order matters. Here, the roles of delegate and alternate mean order is important!

Three research departments have 9, 10, and 6 members, respectively. Each department is to select a delegate and an alternate to represent the department at a conference. In how many ways can this be done?

Explore how to determine the number of ways three departments (with 9, 10, and 6 members) can each select a delegate and an alternate for a conference. This combinatorics problem uses the fundamental counting principle to find the total number of selections. Suitable for high school students.

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