Question

A club wishes to select a president, vice-president and treasurer from five members. How many possible slates of officers are there if no person can hold more than one office?

Short answer

There are 60 possible slates of officers when selecting a president, vice-president, and treasurer from five members without allowing one person to hold more than one office. This is calculated by multiplying the number of ways to choose each position: \(5 \times 4 \times 3 = 60\).

Step-by-step solution

Find the number of ways to select a president

To select a president, we have 5 members to choose from. So there are 5 possible ways to select a president.

Find the number of ways to select a vice-president

Once we have selected a president, we are left with 4 members to choose a vice-president. So there are 4 different ways to select a vice-president.

Find the number of ways to select a treasurer

After selecting a president and vice-president, we have 3 members left to choose a treasurer from. So there are 3 different ways to select a treasurer.

Calculate the total number of ways to select the officers

Since the selections are done without replacement, the total number of ways to select the officers is the product of the number of ways to choose each position. Using the values obtained in Steps 1, 2, and 3, we can calculate the total number of ways as follows: \[ \text{Total number of ways} = 5 \times 4 \times 3 = 60 \] Hence, there are 60 possible slates of officers when selecting a president, vice-president, and treasurer from five members without allowing one person to hold more than one office.

Key concepts

Permutations

Permutations are a fundamental concept in combinatorics, particularly important when the order in which you arrange things matters. In the context of our club's selection of officers, permutations are used because the order of selecting a president, vice-president, and treasurer is important. If you rearrange the same people into different roles, it counts as a different arrangement or permutation.

Permutations can be thought of as rearranging objects. Each distinct order is a unique permutation. When you're selecting officers from a group where no individual can hold more than one position, you look at different ways these individuals can fill the roles, each representing a permutation of the group.

• In our example, choosing from 5 members to fill 3 specific officer roles involves permutations because role assignments create different outcomes.
• Every unique assignment of members to roles is one permutation.
• The calculation involves multiplying the number of choices at each step, reflecting the decreasing pool of available members.

Selection Process

The selection process involves picking candidates for specific roles in a way that respects constraints such as not allowing duplicate assignments to multiple roles. To understand this process, consider it a series of decisions where each choice affects the subsequent ones.

In this problem, the sequence starts with selecting a president from five members. Once chosen, this individual is no longer available for the vice-presidency, leaving four remaining candidates. Next, the vice-president is selected, and subsequently, there are only three people left for the treasurer role.

This method ensures that each position is filled by a different person, maintaining fairness and functionality in the club's leadership structure. It also illustrates the importance of order in selection, as the choice for one role impacts the available options for the others.

• 5 choices are available for the president position.
• 4 remaining choices then exist for the vice-president.
• 3 choices are left for the treasurer, after selections for the other two roles.

Factorial Calculation

Factorials are mathematical operations used in permutations and combinations to find the number of possible arrangements. The notion is based on multiplying a sequence of descending positive numbers. In this context, factorials help calculate the total permutations of objects by considering how many ways you can arrange a given number of elements.

To calculate the number of ways to select officers, you might think about the factorial operation: for a situation with 5 members and 3 roles, a notion similar to factorials is applied. Specifically, though not fully using 5! — which represents counting arrangements of all five members — we use a part of it: 5 × 4 × 3 for our specific needs.

This mirrors the concept of permutations without replacement, capturing how selections decrease as roles are filled one by one. For factorial calculation in broader terms:

• The factorial of a number, denoted as \( n! \), is the product of all positive integers up to \( n \).
• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
• When calculating permutations, we use a form of factorial calculations adapted to the number of selections made.