Softball Team Combinations Problem Solved

[temaire]

Homework Statement

There are three girls and six boys on the school softball team. The team consists of a pitcher, a catcher, four infielders, and three outfielders. How many ways can the nine different positions be filled if the pitcher must be a girl and the catcher must be a boy?

Homework Equations

$$_n{C}_r=\frac{n!}{(n-r)!r!}$$

The Attempt at a Solution

The answer to this problem is $$(_3{C}_1)(_6{C}_1)(7!)$$. However, can someone explain the 7! part?

 

[Tedjn]

The 7! refers to how many ways you can assign the 7 remaining people to the infield and outfield positions after assigning a girl and a boy to be pitcher and catcher.

 

[temaire]

This is what I had thought. However, there are two positions not one, that the rest are assigned to. Doesn't that affect the answer?

 

[Tedjn]

I'm assuming each of the infield and outfield positions are considered different (e.g. left field, right field, center field...)

 

[temaire]

Thanks Ted, that's exactly what I was looking for.