To determine all possible pairs of vice presidents that can be selected from the given group of five (Andrew, Beth, Charles, Diane, and Eric), we need to generate all unique combinations where the order does not matter, and no individual is paired with themselves.
Step-by-step solution:
1. List of vice presidents:
- Andrew
- Beth
- Charles
- Diane
- Eric
2. Determine unique pairs:
- A pair (A, B) is considered the same as (B, A), so we only need to list one of them.
- We will consider pairs (A, B) where A is not equal to B.
3. Generate all possible pairs:
- Starting with Andrew, we pair him with every other vice president:
- (Andrew, Beth)
- (Andrew, Charles)
- (Andrew, Diane)
- (Andrew, Eric)
- Next, we pair Beth with every vice president who comes after her in the list:
- (Beth, Charles)
- (Beth, Diane)
- (Beth, Eric)
- Similarly, continue with Charles:
- (Charles, Diane)
- (Charles, Eric)
- Finally, pair Diane with Eric:
- (Diane, Eric)
4. Complete list of unique pairs:
- (Andrew, Beth)
- (Andrew, Charles)
- (Andrew, Diane)
- (Andrew, Eric)
- (Beth, Charles)
- (Beth, Diane)
- (Beth, Eric)
- (Charles, Diane)
- (Charles, Eric)
- (Diane, Eric)
Listing all the possible pairs, we have:
[tex]\[ \{(Andrew, Beth), (Andrew, Charles), (Andrew, Diane), (Andrew, Eric), (Beth, Charles), (Beth, Diane), (Beth, Eric), (Charles, Diane), (Charles, Eric), (Diane, Eric)\} \][/tex]
Therefore, the correct list of all possible samples of size 2 selected from this population of 5 vice presidents without replacement is:
[tex]\[
(Andrew, Beth), (Andrew, Charles), (Andrew, Diane), (Andrew, Eric), (Beth, Charles), (Beth, Diane), (Beth, Eric), (Charles, Diane), (Charles, Eric), (Diane, Eric)
\][/tex]
