Question

there are ten female board members and twenty male board members. how many ways are there to make a committee of eight board members? how many ways are there to make a committee of eight board members if exactly two must be female? determine the probability of selecting a committee of eight board members where exactly two of the members were female. write your answer as a decimal, rounded to the nearest thousandth.

Explanation:

Step1: Calculate total number of board - members

The total number of board members is $10 + 20=30$.

Step2: Calculate number of ways to form a committee of 8 members

We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 30$ and $r = 8$.
$C(30,8)=\frac{30!}{8!(30 - 8)!}=\frac{30!}{8!×22!}=\frac{30\times29\times28\times27\times26\times25\times24\times23}{8\times7\times6\times5\times4\times3\times2\times1}=5852925$

Step3: Calculate number of ways to choose 2 female members

The number of ways to choose 2 female members out of 10 is $C(10,2)=\frac{10!}{2!(10 - 2)!}=\frac{10\times9}{2\times1}=45$

Step4: Calculate number of ways to choose 6 male members

The number of ways to choose 6 male members out of 20 is $C(20,6)=\frac{20!}{6!(20 - 6)!}=\frac{20\times19\times18\times17\times16\times15}{6\times5\times4\times3\times2\times1}=38760$

Step5: Calculate number of ways to form a committee with 2 female and 6 male members

By the multiplication principle, the number of ways to form a committee of 8 members with 2 female and 6 male members is $C(10,2)\times C(20,6)=45\times38760 = 1744200$

Step6: Calculate the probability

The probability $P$ of forming a committee of 8 members with 2 female members is $P=\frac{C(10,2)\times C(20,6)}{C(30,8)}=\frac{1744200}{5852925}\approx0.298$

Answer:

The number of ways to make a committee of 8 board - members is 5852925.
The number of ways to make a committee of 8 board - members with exactly 2 female members is 1744200.
The probability of selecting a committee of 8 board - members with exactly 2 female members is approximately 0.298.