The first department, with 9 members, has 9 different choices for their delegate and 8 different choices for their alternate --> multiplying them give you 72 different possibilities.
The department with 6 members has 6 different choices for their delegate and 5 different choices for their alternate --> 30 different possibilities.
The third department with 8 members have 8 x 7 = 56 different possibilities.
Since these are independent actions (what one department does has no effect upon what the other departments do), multiply these answers to get a total of 72 x 30 x 56 total possibilities.
