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To solve this problem, let's consider each department individually.

Each department needs to select two people: one delegate and one alternate. Since the selection is specific (one as delegate and another as alternate), the order of choice matters.

1. **First Department (10 members):**  
   - Choose the delegate: There are 10 options.
   - Choose the alternate: Once the delegate is chosen, 9 members remain for the alternate.
   - Total ways for this department = \(10 \times 9 = 90\).

2. **Second Department (7 members):**  
   - Choose the delegate: 7 options.
   - Choose the alternate: 6 remaining options after selecting the delegate.
   - Total ways for this department = \(7 \times 6 = 42\).

3. **Third Department (8 members):**  
   - Choose the delegate: 8 options.
   - Choose the alternate: 7 remaining options.
   - Total ways for this department = \(8 \times 7 = 56\).

Finally, since each department’s selection is independent of the others, we multiply the results for each department:

\[
90 \times 42 \times 56 = 211,680
\]

### Answer:
The total number of ways to select a delegate and an alternate from each department is **211,680**.

Would you like further details on this solution or have any questions?

### Related Questions:
1. How would the calculation change if the departments had different numbers of members?
2. What if each department needed to select three people instead of two?
3. How would we solve this if the same person could be both delegate and alternate?
4. What if each department had to select only one representative without alternates?
5. How does this problem relate to permutations and combinations in combinatorics?

### Tip:
When a problem involves choosing specific roles from a group, consider if the order matters, as this can shift the solution from a combination to a permutation approach.

Three research departments have 10, 7, and 8 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?

Calculate the number of ways to select a delegate and an alternate for each of three research departments with members 10, 7, and 8, respectively, using combinatorics and permutation techniques. Suitable for students in Grades 9-12.

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