Question

there are nine female board members and twenty - one male board members. how many ways are there to make a committee of twelve board members? ways how many ways are there to make a committee of twelve board members if exactly two must be female? ways

Explanation:

Step1: Calculate total number of board - members

The total number of board - members is $9 + 21=30$.

Step2: Calculate number of ways to form a 12 - member committee (using combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$)

We want to choose 12 members out of 30. So $n = 30$ and $r = 12$.
$C(30,12)=\frac{30!}{12!(30 - 12)!}=\frac{30!}{12!×18!}=\frac{30\times29\times\cdots\times19}{12\times11\times\cdots\times1}=86493225$

Step3: Calculate number of ways to choose 2 female members out of 9

We want 2 female members out of 9. Using the combination formula with $n = 9$ and $r = 2$.
$C(9,2)=\frac{9!}{2!(9 - 2)!}=\frac{9!}{2!×7!}=\frac{9\times8}{2\times1}=36$

Step4: Calculate number of ways to choose 10 male members out of 21

We want 10 male members out of 21. Using the combination formula with $n = 21$ and $r = 10$.
$C(21,10)=\frac{21!}{10!(21 - 10)!}=\frac{21!}{10!×11!}=\frac{21\times20\times\cdots\times12}{10\times9\times\cdots\times1}=352716$

Step5: Calculate number of ways to form a 12 - member committee with 2 female members

By the multiplication principle, the number of ways to form a 12 - member committee with exactly 2 female members is the product of the number of ways to choose 2 female members and the number of ways to choose 10 male members.
$C(9,2)\times C(21,10)=36\times352716 = 12697776$

Answer:

The number of ways to make a committee of twelve board - members is 86493225 ways.
The number of ways to make a committee of twelve board - members if exactly two must be female is 12697776 ways.