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.

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.
Transcript text: 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.
failed

Solution

failed
failed

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.

Step 1: Calculate Total Number of Five-Person Committees

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:

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

Step 2: Calculate All-Male Five-Person Committees

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

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

Step 3: Calculate All-Female Five-Person Committees

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

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

Step 4: Calculate Mixed-Gender Five-Person Committees

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

  • Total number of five-person committees: \(\boxed{15504}\)
  • All-male five-person committees: \(\boxed{792}\)
  • All-female five-person committees: \(\boxed{56}\)
  • Mixed-gender five-person committees: \(\boxed{14656}\)
Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Multiple Select Question Select all that apply Which of the following describes the fins of lobe-finned fishes? muscular no muscles fleshy lobe bony rays at the tips
failedAnswered
An X-ray has a wavelength of 1.87 Å. Calculate the energy (in J) of one photon of this radiation. (10^10 Å=1 m). Round your answer to 3 significant figures.
failedAnswered
What are the (y)-intercepts ( f(x)=frac19(x-9)^2+9 )
failedAnswered
For the graph of a function y=f(x) shown to the right, find the absolute maximum and the absolute minimum, if they exist. Identify any local maxima or local minima. Select the correct answer below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute maximum of y=f(x) is f()= . (Type integers or simplified fractions.) B. There is no absolute maximum for y=f(x). Select the correct answer below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute minimum of y=f(x) is f()= (Type integers or simplified fractions.) . B. There is no absolute minimum for y=f(x).
failedAnswered
In a market economy, a significant change in consumers' desire for product X will Multiple Choice alter the profits or losses received by suppliers of product X. cause a reallocation of scarce resources. cause some industries to expand and others to contract. All of these choices are correct.