Step 1

Definition

Product rule If one event can occur in m ways AND a second event can occur in n ways, then the number of ways that the two events can occur in sequence is then \(\displaystyle{m}\cdot{n}\)

Definition permutation (order is important):

\(\displaystyle{P}{\left({n},{r}\right)}={\frac{{{n}!}}{{{\left({n}-{r}\right)}!}}}\)

Definition combination (order is not important):

\(C(n,r)=\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!(n-r)!}\)

\(\displaystyle\text{with }\ {n}\ne{n}\cdot{\left({n}-{1}\right)}\cdot\ldots\cdot{2}\cdot{1}\)

Step 2

Solution

The order of the committee members does not matter (because they all receive the same position), thus we then need to use the definition of combination

(a) Committees with any number of men/women

We need to select 5 members from the 9+7=16 people

n=16

r=5 Evaluate the definition of a combination:

\(\displaystyle{C}{\left({16},{5}\right)}={\frac{{{16}!}}{{{5}!{\left({16}-{5}\right)}!}}}={\frac{{{16}!}}{{{5}!{11}!}}}={4368}\)

Committees with only men

We need to select 5 members from the 9 men

n=9

r=5

Evaluate the definition of a combination:

\(\displaystyle{C}{\left({9},{5}\right)}={\frac{{{9}!}}{{{5}!{\left({9}-{5}\right)}!}}}={\frac{{{9}!}}{{{5}!{4}!}}}={126}\)

Committees with at least one woman

We need to select 5 members from the 9 men

4368-126=4242

Step 3

(b)Committees with only women

We need to select 5 members from the 7 women

n=7

r=5

Evaluate the definition of a combination:

\(\displaystyle{C}{\left({7},{5}\right)}={\frac{{{7}!}}{{{5}!{\left({7}-{5}\right)}!}}}={\frac{{{7}!}}{{{5}!{\left({7}-{5}\right)}!}}}={\frac{{{7}!}}{{{5}!{2}!}}}={21}\)

Committees with at least one woman and at least one man

The possible committees with at least one women are the possible committees that do not have only men and do not have only women:

4368-126-21=4221

Result

(a)4242

(b)4221