Question

Three research departments have 10 ,6 , 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?

Answer 1

Hi !

$\left( \begin{array}{c}10\\2\end{array}\right) \times \left( \begin{array}{c}6\\2\end{array}\right)\times \left( \begin{array}{c}8\\2\end{array}\right) = 18900$

Answer 2

The number of permutations of the number of members in each department, taken two at a time, is needed to solve this problem. In each selection of two members there are two ways to assign the roles of delegate and alternate.
$10P2\times6P2\times8P2=151,200\ ways$