We have been given that three research departments have 6, 9, and 10 members respectively. Each department is to select a delegate and alternate to represent the department at a conference. We are asked to find in how many ways this can be done.
We will solve this problem using permutations because order matters in our selection.
[tex]_{2}^{6}\textrm{P}\cdot_{2}^{9}\textrm{P}\cdot_{2}^{10}\textrm{P}[/tex]
[tex]\frac{6!}{(6-2)!}\cdot\frac{9!}{(9-2)!}\cdot\frac{10!}{(10-2)!}[/tex]
[tex]\frac{6\cdot 5\cdot 4!}{4!}\cdot \frac{9\cdot 8\cdot 7!}{7!}\cdot \frac{10\cdot 9\cdot 8!}{8!}[/tex]
[tex]6\cdot 5\cdot 9\cdot 8\cdot 10\cdot 9=194400[/tex]
Therefore, these selections can be done in 194400 ways.
