Questions: The professor in this class would like to organize a committee to help critique a new textbook. i. How many different five-person committees can be formed? ii. How many different all male five-person committees can be formed? iii. How many different all female five-person committees can be formed? iv. Use your responses to parts (ii) and (iii) above to find how many five-person committees with mixed gender representation can be formed.

Solution Steps

To solve these questions, we need to use combinations, as the order of selection does not matter. We will calculate the total number of five-person committees, then calculate the number of all-male and all-female committees separately. Finally, we will use these results to find the number of mixed-gender committees by subtracting the all-male and all-female committees from the total.

To find the total number of five-person committees that can be formed from 20 people, we use the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

where \( n = 20 \) and \( r = 5 \). Thus, the total number of committees is:

\[ \binom{20}{5} = \frac{20!}{5!(20-5)!} = 15504 \]

To find the number of all-male five-person committees, we use the same combination formula with \( n = 12 \) (the number of males) and \( r = 5 \):

\[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = 792 \]

Similarly, to find the number of all-female five-person committees, we use the combination formula with \( n = 8 \) (the number of females) and \( r = 5 \):

\[ \binom{8}{5} = \frac{8!}{5!(8-5)!} = 56 \]

To find the number of mixed-gender five-person committees, we subtract the number of all-male and all-female committees from the total number of committees:

\[ \text{Mixed-gender committees} = 15504 - 792 - 56 = 14656 \]

Final Answer