SOLUTION: Three research departments have 10 , 7 , and 8 members, respectively. If each department is to select a delegate and an alternate to represent the department at a conference, in ho

Question 1133543: Three research departments have 10 , 7 , and 8 members, respectively. If 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?
Found 2 solutions by Boreal, greenestamps:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
Each department has to select two people but each one can be a delegate or an alternate.
for 10 it is 10C2 or 45, with each one having two possibilities or 90 altogether.
This is clearer if one uses a group of 4
ABCD
AB AC AD BC BD CD
BA CA DA CB BD DC
4C2=6 but there are 12 possiblities
7C2=21 and twice that is 42
8C2=28 and twice that is 56
Total is 188 ways
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Since it is significant whether a member is selected as a delegate or an alternate, there are...

10*9 = 90 ways to choose 2 of the 10;
7*6 = 42 ways to choose 2 of the 7; and
8*7 = 56 ways to choose 2 of the 8.

The total number of ways of selecting a delegate and an alternate from each of the three groups is the PRODUCT of those three numbers:

90*42*56 = 211680
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