A certain company has 30 female employees, including 3 in the management ranks, and 150 male employees, including 12 in the management ranks. A committee consisting of 3 women and 3 men is to be chosen. How many ways are there to choose the committee if:
- It includes at least 1 person of management rank of each gender?
We can pick female employees in the following way: pick any 3 female employees in \(\binom{30}{3} = 4060\) ways. Since \(\binom{27}{3} = 2925\) choices will include 3 non-management rank employees, we need to exclude those choices from the total (so that we get at least one management rank female employee) to get:
\[\binom{30}{3} - \binom{27}{3} = 1135\]
We do the same for male employees:
\[\binom{150}{3} - \binom{138}{3} = 122764\]
The grand total is the product of the number of ways to pick female and male employees:
\[\left(\binom{30}{3} - \binom{27}{3}\right) \cdot \left(\binom{150}{3} - \binom{138}{3}\right) = 139337140\]
- It includes at least 1 person of management rank?
We can use a similar method to the one above. First, let’s pick any 3 female employees and any 3 male employees for the total of:
\[\binom{30}{3} \cdot \binom{150}{3} = 2238278000\]
Many of those choices, however, don’t include any management rank person so we need to subtract such choices from the total. There are a total of:
\[\binom{27}{3} \cdot \binom{138}{3} = 1253467800\] such choices so the final result is:
\[\left(\binom{30}{3} \cdot \binom{150}{3}\right) - \left(\binom{27}{3} \cdot \binom{138}{3}\right) = 984810200\]