Problem 30 - Algebra and Trigonometry

Question

19–32 These problems involve permutations. Class Officers In how many ways can a president, vice president, secretary, and treasurer be chosen from a class of 30 students?

Step-by-Step Solution

Answer

There are 657,720 ways to choose the officers.

Step 1: Identify the Total and Required Positions

We need to select four different positions - a president, vice president, secretary, and treasurer - from a class of 30 students. These roles are distinct, meaning the order in which we choose matters.

Step 2: Determine Permutation Formula

Since order matters, we use the formula for permutations: \[ _nP_r = \frac{n!}{(n-r)!} \]where \(n\) is the total number of items to choose from, and \(r\) is the total number of items to choose.

Step 3: Apply Values to Formula

In this scenario, \(n = 30\) (the total number of students) and \(r = 4\) (the number of positions). Therefore, the calculation becomes: \[ _{30}P_4 = \frac{30!}{(30-4)!} = \frac{30!}{26!} \]

Step 4: Simplify the Calculation

To simplify \(\frac{30!}{26!}\), notice that we can write down only the first four terms of 30 factorial: \[ 30 \times 29 \times 28 \times 27 \times 26! \]Cancel out the \(26!\) in both the numerator and denominator: \[ 30 \times 29 \times 28 \times 27 \]

Step 5: Calculate the Result

Now, calculate the remaining expression: First, compute the product step-by-step:1. \(30 \times 29 = 870\)2. \(870 \times 28 = 24360\)3. \(24360 \times 27 = 657720\)Thus, there are 657720 ways to choose the officers.

Key Concepts

Understanding Factorials

A factorial, represented by the symbol "!", is a function that multiplies a number by every positive integer below it.
For example, the factorial of 4 (written as \(4!\)) is calculated as:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]Factorials are fundamental in permutations and combinations because they help calculate the total number of ways to arrange a set amount of items.
In the context of choosing class officers, we used the factorial to determine how many ways a set number of students can fill specific roles.
If you have 30 students and need to choose one president, you would initially have 30 choices for the president.
Factorials make solving such problems manageable as they provide a systematic way to account for all ordering possibilities.

Permutation Formula Explained

Permutations are about arranging items where the order is significant.
The permutation formula helps compute how many ways you can arrange or select items from a larger pool.
This formula is written as:

• \( _nP_r = \frac{n!}{(n-r)!} \)

It helps find the number of ways to select \(r\) items from \(n\) total items without replacement.
Remember, the order in which you select them matters.
For our class officer scenario, \(n = 30\) students and \(r = 4\) positions to fill.
By applying these to the formula, we calculated the number of ways the students could be chosen for specific roles in the sequence of president, vice president, secretary, and treasurer.

Choosing Class Officers

Now, let's tie it all together with the class officers' selection. We needed to identify how many unique ways the roles of president, vice president, secretary, and treasurer can be assigned among 30 students.
This is a classic permutation problem because each officer role is distinct and the order in which they're chosen matters.

• First, the permutation formula \( _nP_r = \frac{n!}{(n-r)!} \) was applied with \(n = 30\) and \(r = 4\).
• Plugging these values into the permutation formula, we simplified \( \frac{30!}{26!} \).
• The simplification of this expression involved cancelling out the "26!" and multiplying only the top parts: \(30 \times 29 \times 28 \times 27\).

This final calculation revealed there are 657,720 unique ways to assign these officer positions.
Solving such problems with permutations ensures that no potential order or position assignment is overlooked, giving a comprehensive count of all possible outcomes.