Question

The Student Council at a certain school has eight members. Four members will form an executive committee consisting of a president, a vice president, a secretary, and a treasurer.

a) In how many ways can these four positions be filled?
b) In how many ways can four people be chosen for the executive committee if it does not matter who gets which position?
c) Four of the people on Student Council are Zachary, Yolanda, Xavier, and Walter. What is the probability that Zachary is president, Yolanda is vice president, Xavier is secretary, and Walter is treasurer? Round your answers to at least 6 decimal places.
d) What is the probability that Zachary, Yolanda, Xavier, and Walter are the four committee members? Round your answers to at least 6 decimal places.

Answer 1

a) The number of ways to fill the four positions is 1,680. b) The number of ways to choose four members for the executive committee is 70. c) The probability that Zachary is president, Yolanda is vice president, Xavier is secretary, and Walter is treasurer is 0.000595238. d) The probability that Zachary, Yolanda, Xavier, and Walter are the four committee members is 0.000595238.

Step-by-step explanation:

a) The four positions can be filled in a specified order. The first position can be filled by any of the 8 members on the Student Council. After the first position is filled, the second position can be filled by any of the remaining 7 members, and so on. Therefore, the number of ways to fill the four positions is 8 * 7 * 6 * 5 = 1,680.

b) If it does not matter who gets which position, then we do not need to consider the order in which the positions are filled. We can simply count the number of combinations of 4 members from a group of 8. This can be calculated using the formula for combinations, which is 8! / (4!(8-4)!), where ! represents the factorial function. Simplifying this expression gives us 8 * 7 * 6 * 5 / (4 * 3 * 2 * 1) = 70.

c) The probability that Zachary is president, Yolanda is vice president, Xavier is secretary, and Walter is treasurer is the probability of this specific outcome divided by the total number of possible outcomes. The total number of possible outcomes is the same as the number of ways to fill the four positions (1,680). The probability of this specific outcome is 1 / 1,680. Therefore, the probability is 1 / 1,680 = 0.000595238.

d) The probability that Zachary, Yolanda, Xavier, and Walter are the four committee members is the probability of this specific outcome divided by the total number of possible outcomes. The probability of this specific outcome is 1 / (8 * 7 * 6 * 5) = 1 / 1,680. The total number of possible outcomes is the same as the number of ways to fill the four positions (1,680). Therefore, the probability is 1 / 1,680 = 0.000595238.

Answer 2

There are 1680 ways to fill the four executive council positions, 70 ways to choose four people regardless of position, a probability of 0.000595 that four specific individuals will fill specific positions, and a probability of 0.014286 that these four individuals are the chosen committee members.

Step-by-step explanation:

To answer the student's question:

1. The number of ways to fill the four positions of president, vice president, secretary, and treasurer can be found using permutations because the order of selection matters. This is a permutation of 8 items taken 4 at a time, which is calculated as 8! / (8-4)!, and equals 8*7*6*5 = 1680 ways.
2. To choose four people for the executive committee where the positions do not matter, we use combinations. This is a combination of 8 items taken 4 at a time, calculated as 8! / (4!*(8-4)!), and equals 70 ways.
3. To find the probability that Zachary, Yolanda, Xavier, and Walter are each chosen for the specified positions, since there's only one way for this to happen, we divide 1 by the total number of ways to fill the positions (1680), rounding to six decimal places: 0.000595.
4. The probability that these four individuals are the committee members, regardless of position, has to be calculated considering the total ways four people can be chosen out of eight, which results in 1 / 70, rounded to six decimal places: 0.014286.