School parliament

There are 18 boys and 14 girls in the class. In how many ways can three representatives be elected to the school parliament if these are to be:
a) the boys themselves
b) one boy and two girls

Correct answer:

a =  816
b =  1638

Step-by-step explanation:

$$\begin{gathered} C_3(18)=\left(\genfrac{}{}{0ex}{}{18}{3}\right)=\frac{18!}{3!(18-3)!}=\frac{18\cdot 17\cdot 16}{3\cdot 2\cdot 1}=816 \\ a=\left(\genfrac{}{}{0ex}{}{18}{3}\right)=816 \end{gathered}$$

$$\begin{gathered} C_2(14)=\left(\genfrac{}{}{0ex}{}{14}{2}\right)=\frac{14!}{2!(14-2)!}=\frac{14\cdot 13}{2\cdot 1}=91 \\ b=18\cdot \left(\genfrac{}{}{0ex}{}{14}{2}\right)=18\cdot 91=1638 \end{gathered}$$

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