Problem 19 - Algebra and Trigonometry

Question

19–32 These problems involve permutations. Class Officers In how many different ways can a president, vice president, and secretary be chosen from a class of 15 students?

Step-by-Step Solution

Answer

There are 2730 different ways.

Step 1: Identify the Problem Type

This exercise is about finding the number of permutations. A permutation is an arrangement of objects in a specific order. For the class officers, we are choosing and arranging 3 positions: president, vice president, and secretary from 15 students.

Step 2: Use the Permutation Formula

The formula for permutations is given by \( nPr = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 15 \) and \( r = 3 \).

Step 3: Plug in the Values

Substitute \( n = 15 \) and \( r = 3 \) into the permutation formula: \[ 15P3 = \frac{15!}{(15-3)!} = \frac{15!}{12!} \]

Step 4: Simplify the Expression

Cancel the \( 12! \) from the numerator and denominator:\[ 15P3 = 15 \times 14 \times 13 \]This eliminates the factorial of 12 since everything below 13 in the numerator and denominator cancels out.

Step 5: Compute the Result

Calculate the product:\[ 15 \times 14 = 210 \]\[ 210 \times 13 = 2730 \]That gives the total number of ways to choose the officers.

Key Concepts

Understanding Combination Formula

While the term **combination formula** is often used, it's important to understand its role. In mathematics, a combination is a way of selecting items from a larger pool, where the order does not matter. This is different from permutations, where the order is crucial. The formula used for combinations is different from permutations:

• For combinations, the formula is \( nCr = \frac{n!}{r!(n-r)!} \).
• Here, \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, but without regard for the order.

If you were organizing a committee and it didn't matter who exactly was chosen for each role, you'd use combinations. However, for this class officer problem, where the roles of president, vice president, and secretary are specific, the permutation formula is more appropriate.

Exploring Factorial

The term **factorial**, represented by an exclamation mark (!), is a cornerstone of permutations and combinations. A factorial, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Factorials play a critical role in many mathematical formulas because they help calculate how many different ways a set of items can be arranged. This operation simplifies complex calculations by allowing for cancelations, like in the permutation formula where terms in the numerator and the denominator cancel each other to simplify the expression.

Importance of Arrangement

Understanding **arrangement** is key when diving into permutations. In permutations, the order of the selected items matters significantly. This means that different orders of the same items are treated as unique arrangements. For the class officer problem, each specific position—president, vice president, and secretary—creates a different arrangement of students. Let's look at a simple example:

• If we have students A, B, and C and the order is important, then 'A as president, B as vice president, C as secretary' is completely different from 'C as president, A as vice president, B as secretary'.
• Thus different arrangements need to be calculated, hence why permutations are used over combinations.

So, when asked for permutations, always look at the distinct roles or requirements involved.

Applying Mathematical Problem Solving

**Mathematical problem solving** involves understanding the problem deeply, identifying the right approach, and executing the solution efficiently. Breaking down the problem can make it easier to solve:

• Identify the situation: Determine if a permutation or combination approach is necessary. Here, it's permutations because the position's order matters.
• Understand the formula: Use the appropriate formula, in this case, \( nPr = \frac{n!}{(n-r)!} \).
• Perform calculations: Substitute the correct values into the equations and simplify step by step as shown in the original solution, ensuring a clear path to the answer.

This step-by-step breakdown helps avoid errors and checks comprehension of every part of the problem, leading to the correct outcome, as demonstrated in the class officer permutation exercise.