# Combinations Problem

## 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?

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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.

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?

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

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