Problem 35 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(7,3)\)

Step-by-Step Solution

Answer

The value of \(P(7, 3)\) is 210.

Step 1: Understand the Permutation Formula

To evaluate the permutation, use the formula for permutations: \(P(n, r) = \frac{n!}{(n-r)!}\). In this case, \(n = 7\) and \(r = 3\). This formula calculates the number of ways to arrange \(r\) objects out of \(n\) objects.

Step 2: Substitute Values into the Formula

Substitute the values of \(n\) and \(r\) into the permutation formula: \(P(7, 3) = \frac{7!}{(7-3)!}\). This simplifies to \(P(7, 3) = \frac{7!}{4!}\).

Step 3: Calculate the Factorials

Calculate the factorials involved: \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) and \(4! = 4 \times 3 \times 2 \times 1\). Therefore, \(7! = 5040\) and \(4! = 24\).

Step 4: Divide the Factorials

Now divide the factorial of 7 by the factorial of 4: \(\frac{7!}{4!} = \frac{5040}{24}\).

Step 5: Calculate the Final Result

Perform the division: \(\frac{5040}{24} = 210\). Therefore, \(P(7, 3) = 210\).

Key Concepts

Factorials

Understanding factorials is essential when dealing with permutations and combinations in mathematics. Factorials are mathematical operations that involve multiplying a series of descending natural numbers from a given number down to one. The notation for a factorial is an exclamation mark (!). For example, 7 factorial is written as \(7!\) and calculates as: - 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 This results in 5040. Factorials are significant in various areas of combinatorial mathematics because they help calculate the total number of ways to arrange objects or determine orderings systematically. In permutations, factorials are used to find how many different sequences can be formed from a set of objects. Whenever you see an expression like \(P(n, r)\), rest assured that factorials play a pivotal role in simplifying and solving the equation.

It's important to note that 0! is defined as 1, which serves as a foundation for many mathematical principles involving factorials.

Combinatorial Mathematics

Combinatorial mathematics is a fascinating field that revolves around counting and arranging objects. It plays a crucial role in determining how objects can be organized in certain ways. This branch of mathematics encompasses concepts such as permutations, combinations, and binomial coefficients.

The primary focus of combinatorial mathematics is to understand the different combinations and permutations that can be formed with a set of elements, and this includes calculating probabilities. Permutations, like \(P(n, r)\), specifically deal with the arrangement of objects in which the order does matter.

• When calculating permutations, it often involves factorials to determine how elements can be ordered.
• Combinatorial problems can range from simple to complex, frequently requiring logical reasoning and formula application.

Combinatorics has applications in computer science, statistics, and many areas where problem-solving with discrete structures is necessary.

Arrangement of Objects

Arranging objects, also known as permutations, is about organizing a set of items in sequences where the order is significant. For example, if you have 3 books and you want to arrange them on a shelf, you can place them in different orders. The formula for permutations \(P(n, r)\) finds the number of possible arrangements where \(n\) is the total number of items to choose from, and \(r\) is the number of items to arrange.

In our example, \(P(7, 3)\) means arranging 3 items from a total of 7. The process involved:

• Calculating factorials, which provide the basis for determining possible arrangements.
• Utilizing the formula \(\frac{n!}{(n-r)!}\) to compute the possible sequences.

This results in the number 210, showing that there are 210 different ways to arrange 3 out of 7 objects. Understanding permutations is critical in scenarios where the sequence or order of selection matters, such as in scheduling, coding, and cryptography.