Solved on Dec 13, 2023

A club $\mathrm{N}$ with 5 members. Elect a president and treasurer such that no one holds more than one office. How many ways?
A. $\mathrm{AA}, \mathrm{BB}, \mathrm{CC}, \mathrm{DD}, \mathrm{EE}$ B. $\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}, \mathrm{CD}, \mathrm{CE}, \mathrm{DE}$ C. $\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}, \mathrm{BA}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}, \mathrm{CA}, \mathrm{CB}, \mathrm{CD}, \mathrm{CE}, \mathrm{DA}, \mathrm{DB}, \mathrm{DC}, \mathrm{DE}, \mathrm{EA}, \mathrm{EB}, \mathrm{EC}, \mathrm{ED}$ D. $\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}, \mathrm{BA}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}, \mathrm{CA}, \mathrm{CB}, \mathrm{CD}, \mathrm{CE}, \mathrm{DA}, \mathrm{DB}, \mathrm{DC}, \mathrm{DE}, \mathrm{EA}, \mathrm{EB}, \mathrm{EC}, \mathrm{ED}$

STEP 1

Assumptions
1. The club $\mathrm{N}$ has five members: Anthony, Blake, Carrie, Dennis, Elaine, abbreviated as $\mathrm{N}=\{\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\}$ .
2. Two distinct offices are to be filled: president and treasurer.
3. No member can hold more than one office.
4. All members are eligible for both offices.

STEP 2

To find the different ways the club could elect a president and a treasurer, we first select a president. Since there are five members, there are five choices for president.

STEP 3

After electing a president, we need to elect a treasurer. Since the president cannot be the treasurer, we have four remaining choices for treasurer.

STEP 4

The total number of ways to elect a president and a treasurer is the product of the number of choices for president and the number of choices for treasurer.
$Total\, ways = Choices\, for\, president \times Choices\, for\, treasurer$

STEP 5

Plug in the values for the number of choices for president (5) and treasurer (4) to calculate the total number of ways.
$Total\, ways = 5 \times 4$

STEP 6

Calculate the total number of ways to elect a president and a treasurer.
$Total\, ways = 5 \times 4 = 20$

STEP 7

Now, we need to list the possible pairs of presidents and treasurers. We will create pairs where the first member of the pair is the president and the second member is the treasurer.

STEP 8

List all possible pairs where Anthony (A) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}\}$

STEP 9

List all possible pairs where Blake (B) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{BA}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}\}$

STEP 10

List all possible pairs where Carrie (C) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{CA}, \mathrm{CB}, \mathrm{CD}, \mathrm{CE}\}$

STEP 11

List all possible pairs where Dennis (D) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{DA}, \mathrm{DB}, \mathrm{DC}, \mathrm{DE}\}$

STEP 12

List all possible pairs where Elaine (E) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{EA}, \mathrm{EB}, \mathrm{EC}, \mathrm{ED}\}$

STEP 13

Combine all the pairs listed in steps 8 to 12 to get the complete list of possible pairs of presidents and treasurers.
$\mathrm{Pairs} = \{\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}, \mathrm{BA}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}, \mathrm{CA}, \mathrm{CB}, \mathrm{CD}, \mathrm{CE}, \mathrm{DA}, \mathrm{DB}, \mathrm{DC}, \mathrm{DE}, \mathrm{EA}, \mathrm{EB}, \mathrm{EC}, \mathrm{ED}\}$

STEP 14

The correct list of possible pairs of presidents and treasurers is the one that includes all unique combinations where no member is both president and treasurer, which matches option D.
$\mathrm{Correct\, List} = \mathrm{D}$

STEP 15

The number of ways a president and treasurer can be elected is 20.
The correct answer is option D and there are 20 ways to elect a president and a treasurer.

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STEP 1

STEP 2

To find the different ways the club could elect a president and a treasurer, we first select a president. Since there are five members, there are five choices for president.

STEP 3

After electing a president, we need to elect a treasurer. Since the president cannot be the treasurer, we have four remaining choices for treasurer.

STEP 4

The total number of ways to elect a president and a treasurer is the product of the number of choices for president and the number of choices for treasurer.
$Total\, ways = Choices\, for\, president \times Choices\, for\, treasurer$

STEP 5

Plug in the values for the number of choices for president (5) and treasurer (4) to calculate the total number of ways.
$Total\, ways = 5 \times 4$

STEP 6

Calculate the total number of ways to elect a president and a treasurer.
$Total\, ways = 5 \times 4 = 20$

STEP 7

Now, we need to list the possible pairs of presidents and treasurers. We will create pairs where the first member of the pair is the president and the second member is the treasurer.

STEP 8

List all possible pairs where Anthony (A) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}\}$

STEP 9

List all possible pairs where Blake (B) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{BA}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}\}$

STEP 10

List all possible pairs where Carrie (C) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{CA}, \mathrm{CB}, \mathrm{CD}, \mathrm{CE}\}$

STEP 11

List all possible pairs where Dennis (D) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{DA}, \mathrm{DB}, \mathrm{DC}, \mathrm{DE}\}$

STEP 12

List all possible pairs where Elaine (E) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{EA}, \mathrm{EB}, \mathrm{EC}, \mathrm{ED}\}$

STEP 13

STEP 14

The correct list of possible pairs of presidents and treasurers is the one that includes all unique combinations where no member is both president and treasurer, which matches option D.
$\mathrm{Correct\, List} = \mathrm{D}$

STEP 15

The number of ways a president and treasurer can be elected is 20.
The correct answer is option D and there are 20 ways to elect a president and a treasurer.

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STEP 1

STEP 2

To find the different ways the club could elect a president and a treasurer, we first select a president. Since there are five members, there are five choices for president.

STEP 3

After electing a president, we need to elect a treasurer. Since the president cannot be the treasurer, we have four remaining choices for treasurer.

STEP 4

STEP 5

Plug in the values for the number of choices for president (5) and treasurer (4) to calculate the total number of ways.Total ways=5×4Total\, ways = 5 \times 4Totalways=5×4

STEP 6

Calculate the total number of ways to elect a president and a treasurer.Total ways=5×4=20Total\, ways = 5 \times 4 = 20Totalways=5×4=20

STEP 7

Now, we need to list the possible pairs of presidents and treasurers. We will create pairs where the first member of the pair is the president and the second member is the treasurer.

STEP 8

List all possible pairs where Anthony (A) is the president and someone else is the treasurer.Pairs={AB,AC,AD,AE}\mathrm{Pairs} = \{\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}\}Pairs={AB,AC,AD,AE}

STEP 9

List all possible pairs where Blake (B) is the president and someone else is the treasurer.Pairs={Pairs,BA,BC,BD,BE}\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{BA}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}\}Pairs={Pairs,BA,BC,BD,BE}

STEP 10

List all possible pairs where Carrie (C) is the president and someone else is the treasurer.Pairs={Pairs,CA,CB,CD,CE}\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{CA}, \mathrm{CB}, \mathrm{CD}, \mathrm{CE}\}Pairs={Pairs,CA,CB,CD,CE}

STEP 11

List all possible pairs where Dennis (D) is the president and someone else is the treasurer.Pairs={Pairs,DA,DB,DC,DE}\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{DA}, \mathrm{DB}, \mathrm{DC}, \mathrm{DE}\}Pairs={Pairs,DA,DB,DC,DE}

STEP 12

List all possible pairs where Elaine (E) is the president and someone else is the treasurer.Pairs={Pairs,EA,EB,EC,ED}\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{EA}, \mathrm{EB}, \mathrm{EC}, \mathrm{ED}\}Pairs={Pairs,EA,EB,EC,ED}

STEP 13

STEP 14

The correct list of possible pairs of presidents and treasurers is the one that includes all unique combinations where no member is both president and treasurer, which matches option D.Correct List=D\mathrm{Correct\, List} = \mathrm{D}CorrectList=D

STEP 15

The number of ways a president and treasurer can be elected is 20.The correct answer is option D and there are 20 ways to elect a president and a treasurer.

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Plug in the values for the number of choices for president (5) and treasurer (4) to calculate the total number of ways.
$Total\, ways = 5 \times 4$

Calculate the total number of ways to elect a president and a treasurer.
$Total\, ways = 5 \times 4 = 20$

List all possible pairs where Anthony (A) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{AB}, \mathrm{AC}, \mathrm{AD}, \mathrm{AE}\}$

List all possible pairs where Blake (B) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{BA}, \mathrm{BC}, \mathrm{BD}, \mathrm{BE}\}$

List all possible pairs where Carrie (C) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{CA}, \mathrm{CB}, \mathrm{CD}, \mathrm{CE}\}$

List all possible pairs where Dennis (D) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{DA}, \mathrm{DB}, \mathrm{DC}, \mathrm{DE}\}$

List all possible pairs where Elaine (E) is the president and someone else is the treasurer.
$\mathrm{Pairs} = \{\mathrm{Pairs}, \mathrm{EA}, \mathrm{EB}, \mathrm{EC}, \mathrm{ED}\}$

The correct list of possible pairs of presidents and treasurers is the one that includes all unique combinations where no member is both president and treasurer, which matches option D.
$\mathrm{Correct\, List} = \mathrm{D}$

The number of ways a president and treasurer can be elected is 20.
The correct answer is option D and there are 20 ways to elect a president and a treasurer.