A club $\mathrm{N}$ with five members is shown below.
$N=\{$ Alvin, Bob, Carla, Donald, Emma $\}$, abbreviated as $N=\{A, B, C, D, E\}$
Assuming all members of the club are eligible, but that no one can hold more than one office, list and count the different ways the club could elect both a president and a treasurer.
Choose the correct list of possible pairs of presidents and treasurers.
A. AA, BB, CC, DD, EE
B. $A B, A C, A D, A E, B A, B C, B D, B E, C A, C B, C D, C E, D A, D B, D C, D E, E A, E B, E C, E D$
c. $A B, A C, A D, A E, B C, B D, B E, C D, C E, D E$
D. $A A, A B, A C, A D, A E, B A, B B, B C, B D, B E, C A, C B, C C, C D, C E, D A, D B, D C$, $D D, D E, E A, E B, E C, E D, E E$
How many ways can a president and treasurer be elected?

Step 1 ：The problem is asking for the number of ways to choose a president and a treasurer from a group of 5 people, where no one can hold more than one office. This means that the same person cannot be both the president and the treasurer.

Step 2 ：Therefore, for each choice of president, there are 4 remaining choices for treasurer. Since there are 5 choices for president, the total number of ways to choose a president and a treasurer is \(5 \times 4 = 20\).

Step 3 ：The correct list of possible pairs of presidents and treasurers is option B, because it includes all possible pairs where the president and treasurer are different people.

Step 4 ：Final Answer: The correct list of possible pairs of presidents and treasurers is option B. The number of ways a president and treasurer can be elected is \(\boxed{20}\).