Problem 37 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(25,2)\)

Step-by-Step Solution

Answer

The value of \(P(25, 2)\) is 600.

Step 1: Understand the Definition of Permutation

Permutation refers to the arrangement of objects in a specific order. The notation \(P(n, r)\) represents the number of permutations of \(n\) distinct objects taken \(r\) at a time.

Step 2: Set Up the Permutation Formula

The formula for permutations is \(P(n, r) = \frac{n!}{(n-r)!}\). Here, \(n = 25\) and \(r = 2\). Substitute these values into the formula to get \(P(25, 2) = \frac{25!}{(25-2)!}\).

Step 3: Simplify the Denominator

Calculate \((n-r)!\), which is \((25-2)! = 23!\). Now the expression becomes \(P(25, 2) = \frac{25!}{23!}\).

Step 4: Expand and Simplify the Factorials

The factorial \(25!\) is the product of all natural numbers up to 25. When divided by \(23!\), the expression simplifies to \(25 \times 24\) because all terms in \(23!\) cancel out. So, \(P(25, 2) = 25 \times 24\).

Step 5: Perform the Multiplication

Calculate the product: \(25 \times 24 = 600\). This is the number of ways to arrange 2 objects out of 25 in sequence.

Key Concepts

Factorial

A factorial, denoted by an exclamation point after a number, is a mathematical operation used to calculate the product of an integer and all the integers below it, down to 1. For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\). It is important to note that by definition, \(0!\) is equal to 1.

Factorials are commonly used in permutations and combinations, where they play a crucial role in determining how many ways we can arrange or select items.

• To calculate \(n!\), begin with the number \(n\) and multiply it by each consecutive integer below it until you reach 1.
• Factorials grow very quickly, so they can result in extremely large numbers, even for relatively small values of \(n\).

Permutation Formula

In mathematics, a permutation is an arrangement of objects in a specific order, and the permutation formula helps to determine the number of such possible arrangements.

The permutation formula is expressed as \(P(n, r) = \frac{n!}{(n-r)!}\), where \(n\) is the total number of objects to choose from, and \(r\) is the number of objects to arrange. The formula calculates how many ways you can arrange \(r\) objects out of a total of \(n\) objects.

• The formula uses factorials to account for the arrangements of both selected and unselected objects.
• The denominator \((n-r)!\) helps to exclude the arrangements of the unselected objects, effectively counting only the desired arrangements.

Understand that permutations differ from combinations, as permutations consider order, while combinations do not. This distinction is vital in calculations involving sequences or arrangements.

Combinatorics

Combinatorics is a branch of mathematics dealing with the study of finite or discrete structures. It involves counting, arranging, and analyzing the combinations and permutations of sets of elements. This field is essential for understanding how objects can be arranged or grouped in various scenarios.

Combinatorics includes several key concepts:

• Permutations: These are arrangements of objects where the order matters. For example, different ways to line up people or arrange books on a shelf.
• Combinations: These involve selecting objects where the order does not matter. For instance, choosing ingredients for a recipe without caring about the order of selection.
• Factorials: As mentioned earlier, they are crucial for calculating permutations and combinations.

These concepts are widely used in fields such as computer science, probability theory, and statistics, making combinatorics a fundamental area of study for students encountering mathematical arrangements.