Problem 32 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(10,2)\)

Step-by-Step Solution

Answer

The value of \( P(10, 2) \) is 90.

Step 1: Understand the Concept of Permutation

Permutations are the different ways in which a set or number of things can be arranged. The notation \( P(n, r) \) represents the number of permutations of \( n \) items taken \( r \) at a time. It is important to remember that the order matters in permutations.

Step 2: Recall the Permutation Formula

The formula for permutations \( P(n, r) \) is given by \[ P(n, r) = \frac{n!}{(n-r)!} \]. In this formula, \(!\) denotes factorial, which is the product of all positive integers up to that number. This formula helps us calculate the number of possible arrangements.

Step 3: Substitute Values into the Formula

Substitute \( n = 10 \) and \( r = 2 \) into the permutation formula: \[ P(10, 2) = \frac{10!}{(10-2)!} \]. This simplifies to \[ P(10, 2) = \frac{10!}{8!} \].

Step 4: Calculate the Factorials

Evaluate the factorials \( 10! \) and \( 8! \). - \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)Thus, \( 10! = 3628800 \) and \( 8! = 40320 \).

Step 5: Simplify the Expression

Divide \( 10! \) by \( 8! \): \[ P(10, 2) = \frac{10!}{8!} = \frac{3628800}{40320} \]. To simplify this calculation, cancel out the common terms: \( P(10, 2) = 10 \times 9 = 90 \).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially when it comes to permutations and combinations. A factorial of a non-negative integer is the product of all positive integers less than or equal to that number. Represented as \(!n!\), the factorial of a number signifies the number of ways to arrange \( n \) distinct objects. Factorials grow very rapidly with the increase of \( n \). For instance:

• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

In the context of permutations, we use factorials to understand how many different arrangements are possible for a selected subset.
For example, in our exercise, we used \( 10! \) and \(!8!\) to calculate permutations by canceling common terms. This method simplifies the computation significantly.

Permutation Formula

The Permutation Formula is a technique used in probability and statistics to determine the number of possible arrangements for a set of items, where order matters. This is crucial in scenarios where the sequence or order of selection affects the outcome. The formula itself is expressed as:
\[ P(n, r) = \frac{n!}{(n-r)!} \]Here:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to arrange.
• The exclamation mark \(!\) indicates a factorial.

This formula provides us a way to comprehend the arrangement possibilities given a set of constraints. For instance, calculating \( P(10, 2) \) as shown in the exercise helps find how many different ways two items can be arranged out of ten.

Counting Techniques

Counting Techniques encompass various mathematical methods to calculate the number of possible arrangements or selections. In the world of permutations and combinations, counting principles enable a streamlined approach to solving problems. Two primary principles often used are:

• **Multiplication Principle**: If an event A can occur in \( m \) ways and an event B can occur in \( n \) ways, then the events occurring together can happen in \( m \times n \) ways.
• **Addition Principle**: If an event A can occur in \( m \) ways and an event B can occur in \( n \) ways, without any overlap in occurrences, then the number of ways either A or B can occur is \( m + n\).

Permutations specifically make use of these principles to calculate the number of arrangements where order is significant. Using the permutation example \( P(10, 2) = 10 \times 9 = 90 \), showcases how few selected items can be arranged in multiple ways using these counting techniques.