Three research departments have 6, 10, and 9 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?

For the first department (with 6 people), we have 6 possibilities for the delegate, and then a remaining 5 for the alternate, so that's 6*5 total combinations. Use the same logic for the departments with 10 and 9 people.

The total number of combinations is therefore 6*5*10*9*9*8= 194400 ways