Answer
$252$ different starting lineups.
Work Step by Step
- First, let's select Payton or Jordyn for goalie. We have two choices.
$N(goalie)=2$
- Now, we need to select 4 fielders among the 9 remaining players:
1) It is not a sequence of choices.
2) The order in which the players are selected does not matter.
It is a combination of 9 distinct players taken 4 at a time:
$_{9}C_4=\frac{9!}{4!(9-4)!}=\frac{9!}{4!\times5!}=\frac{9\times8\times7\times6\times5\times4\times3\times2\times1}{4\times3\times2\times1\times5\times4\times3\times2\times1}=126$
Using the Multiplication Rule of Counting (page 298):
$N(starting~lineups)=2\times126=252$
