# 2008 iTest Problems/Problem 27

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## Problem

Hannah Kubik leads a local volunteer group of thirteen adults that takes turns holding classes for patients at the Children’s Hospital. At the end of August, Hannah took a tour of the hospital and talked with some members of the staﬀ. Dr. Yang told Hannah that it looked like there would be more girls than boys in the hospital during September. The next day Hannah brought the volunteers together and it was decided that three women and two men would volunteer to run the September classes at the Children’s Hospital. If there are exactly six women in the volunteer group, how many combinations of three women and two men could Hannah choose from the volunteer group to run the classes?

## Solution

Note that it does not matter what order the people are getting picked, so combinations should be used. There are $\tbinom{6}{3} = 20$ possible ways to choose women and $\tbinom{7}{2} = 21$ possible ways to choose men. Since the two scenarios are independent, there are a total of $20 \cdot 21 = \boxed{420}$ ways for Hannah to choose three women and two men to run the classes.