A department

There are seven women and five men in a department.
a) how many ways can a committee of 3 people be selected?
b) how many ways can a committee of 2 men and one woman be selected?
c) how many ways can a committee of at least two women be selected (3 people total)?

Correct answer:

a =  220
b =  70
c =  140

Step-by-step explanation:

$$\begin{gathered} w=7 \\ m=5 \\ t=w+m=7+5=12 \\ a=\frac{t\cdot (t-1)\cdot (t-2)}{3!}=\frac{12\cdot (12-1)\cdot (12-2)}{3!}=220 \end{gathered}$$

$b=m\cdot (m-1)/2\cdot w=5\cdot (5-1)/2\cdot 7=70$

$$\begin{gathered} C_2(7)=\left(\genfrac{}{}{0ex}{}{7}{2}\right)=\frac{7!}{2!(7-2)!}=\frac{7\cdot 6}{2\cdot 1}=21 \\ {C}_3(7)=\left(\genfrac{}{}{0ex}{}{7}{3}\right)=\frac{7!}{3!(7-3)!}=\frac{7\cdot 6\cdot 5}{3\cdot 2\cdot 1}=35 \\ c=\left(\genfrac{}{}{0ex}{}{w}{2}\right)\cdot m+\left(\genfrac{}{}{0ex}{}{w}{3}\right)=21\cdot 5+35=140 \end{gathered}$$

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