Problem 6 - Algebra and Trigonometry

Question

Evaluate the expression. $$ P(9,2) $$

Step-by-Step Solution

Answer

The number of permutations \( P(9,2) \) is 72.

Step 1: Understanding the Permutation Notation

The notation \( P(n,r) \) represents the number of ways to arrange \( r \) objects from a set of \( n \) distinct objects. This is known as a permutation.

Step 2: Apply the Permutation Formula

The formula for permutations is given by \( P(n,r) = \frac{n!}{(n-r)!} \). Let's apply the formula to our problem, where \( n = 9 \) and \( r = 2 \).

Step 3: Calculate the Factorials

We need to calculate two factorials: \( 9! \) and \( (9-2)! = 7! \). Calculate these values:- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)

Step 4: Simplify the Expression

Simplify \( \frac{9!}{7!} \) by canceling out the common \( 7! \) factorial:\[ \frac{9 \times 8 \times 7!}{7!} = 9 \times 8 = 72 \]

Step 5: Conclude the Calculation

The calculation shows that \( P(9,2) = 72 \). This means there are 72 different ways to arrange 2 objects from a set of 9 distinct objects.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially in combinatorics and permutations. The factorial of a number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It is used to calculate the number of ways to arrange a set of items. For example:

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

Factorials grow very quickly as \( n \) increases, and this rapid increase is useful to calculate arrangements of large sets. In permutations, factorials help simplify expressions when calculating the number of ways to order a specific number of elements.

Permutation Formula

Permutations refer to the different ways of arranging a set of items. The permutation formula, \( P(n, r) = \frac{n!}{(n-r)!} \), helps us determine how many possible arrangements can be made from a set of \( n \) items taking \( r \) at a time.
Here’s how it works:

• \( n \): Represents the total number of items in the set.
• \( r \): The number of items to arrange or select.
• The formula considers all possible sequences of the selected items.

For example, for \( P(9, 2) \), there are 9 items total, and we want to find out the number of ways to arrange 2 of them. Using the formula, it simplifies to calculating \( \frac{9!}{7!} = 9 \times 8 = 72 \). So, there are 72 possible permutations of 2 items from 9.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It plays a crucial role in determining how items can be organized or selected under certain constraints.

• Permutations: Concerned with the order of arrangements. The order in which objects appear matters.
• Combinations: Unlike permutations, combinations are concerned with the selection of objects irrespective of order.

Understanding these concepts helps solve problems where the arrangement, selection, or grouping of items is key. In P(9,2), we're utilizing combinatorics to determine how many different ways we can arrange two items selected from nine distinct items, where the order does matter. This forms the basis of many real-world applications such as scheduling, organizing tournaments, or seating arrangements.