Choosing people

Question

A class has 5 boys and 7 girls. Find the number of ways to choose a team of 4 to represent the class if

1. there are no restrictions,
2. the team is made up of 3 boys and 1 girl,
3. the team must have at least 3 boys
4. the team must have at least one boy

Solution

1. Number of ways $=\left(\genfrac{}{}{0px}{}{12}{4}\right)=495∎$
2. Number of ways $=\left(\genfrac{}{}{0px}{}{5}{3}\right)\times \left(\genfrac{}{}{0px}{}{7}{1}\right)=70∎$

Case 1: 3 boys and 1 girl
Case 2: 4 boys and 0 girls

Number of ways $=\left(\genfrac{}{}{0px}{}{5}{3}\right)\times \left(\genfrac{}{}{0px}{}{7}{1}\right)+\left(\genfrac{}{}{0px}{}{5}{4}\right)=70+5=75∎$

We consider the complement, which is that there are no boys (the entire team of 4 is made up of girls)

Number of ways $=\left(\genfrac{}{}{0px}{}{12}{4}\right)-\left(\genfrac{}{}{0px}{}{7}{4}\right)=495-35=460∎$