Question

a manufacturing company has 5 vice presidents: andrew, beth, charles, diane, and eric. their regional responsibilities are shown in the table. the president of the company wants to select 2 of the 5 vice presidents randomly to send to a conference. which of the following gives a correct list of all possible samples of size 2 selected from this population of 5 vice presidents without replacement? (andrew, beth), (charles, diane), (beth, charles), (diane, eric) (andrew, beth), (andrew, charles), (andrew, diane), (andrew, eric) (andrew, beth), (andrew, charles), (andrew, diane), (andrew, eric), (beth, charles), (beth, diane), (beth, eric), (charles, diane), (charles, eric), (diane, eric) (andrew, andrew), (andrew, beth), (andrew, charles), (andrew, diane), (andrew, eric), (beth, beth), (beth, charles), (beth, diane), (beth, eric), (charles, charles), (charles, diane), (charles, eric), (diane, diane) (diane, eric)

Explanation:

To find all possible samples of size \(2\) selected from \(5\) vice - presidents (Andrew, Beth, Charles, Diane, Eric) without replacement, we use the combination concept. The number of combinations \(C(n,k)=\frac{n!}{k!(n - k)!}\), where \(n = 5\) and \(k=2\), \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10\).
We list all pairs:

• \((Andrew,Beth)\)
• \((Andrew,Charles)\)
• \((Andrew,Diane)\)
• \((Andrew,Eric)\)
• \((Beth,Charles)\)
• \((Beth,Diane)\)
• \((Beth,Eric)\)
• \((Charles,Diane)\)
• \((Charles,Eric)\)
• \((Diane,Eric)\)

The first option has only \(4\) pairs, the second option has only \(4\) pairs, and the fourth option has pairs with the same person (e.g., \((Andrew,Andrew)\)) which is not allowed as we are selecting without replacement.

Answer:

\((Andrew, Beth), (Andrew, Charles), (Andrew, Diane), (Andrew, Eric), (Beth, Charles), (Beth, Diane), (Beth, Eric), (Charles, Diane), (Charles, Eric), (Diane, Eric)\) (corresponding to the third option in the given choices)