Clarice, Dominique, John, and Marco work for a publishing company. The company wants to send two employees to a statistics conference. To be fair, the company decides that the two individuals who get to attend will have their names randomly drawn from a hat.
(a) Determine the sample space of the experiment. That is, list all possible simple random samples of size $n=2$.
(b) What is the probability that Clarice and John attend the conference?
(c) What is the probability that Clarice attends the conference?
(d) What is the probability that John stays home?
(a) Choose the correct answer below. Note that each person is represented by the first letter in their name.
A. CD, CJ, CM, DJ, DM, JM, DC, JC, MC, JD, MD, MJ
B. $\mathrm{CD}, \mathrm{CJ}, \mathrm{CM}$
C. $\mathrm{CD}, \mathrm{CJ}, \mathrm{CM}, \mathrm{DJ}, \mathrm{DM}, \mathrm{JM}$
D. $\mathrm{CD}, \mathrm{CJ}, \mathrm{CM}, \mathrm{DJ}, \mathrm{DM}, \mathrm{JM}, \mathrm{CC}, \mathrm{DD}, \mathrm{JJ}, \mathrm{MM}$
Answer
Final Answer: The correct answer is \(\boxed{C}\).
Step 1 ：The question is asking for all possible combinations of 2 employees from a group of 4. This is a combination problem, not a permutation problem, because the order in which the employees are chosen does not matter. Therefore, we can use the combination formula to calculate the number of combinations. The combination formula is \(\frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose. In this case, \(n=4\) and \(r=2\).
Step 2 ：The calculation gives us 6 possible combinations of 2 employees from a group of 4. This means that the sample space of the experiment consists of 6 possible outcomes. Looking at the answer choices, we can see that option C lists 6 combinations, while the other options list either more or less than 6. Therefore, the correct answer is option C.
Step 3 ：Final Answer: The correct answer is \(\boxed{C}\).
