Problem 2 - College Algebra

Question

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{7} P_{3} $$

Step-by-Step Solution

Answer

The value of \(_7P_3\) is 210.

Step 1: Identify the values of n and r

From the notation \(_7P_3\), we can clearly see that \(n=7\) and \(r=3\).

Step 2: Apply the Permutation Formula

The formula for calculating permutation is \(_nP_r = \frac{n!}{(n-r)!}\). Substituting our values into the formula, we get \(_7P_3 = \frac{7!}{(7-3)!}\).

Step 3: Calculate the Factorials and Solve

7! means the product of all positive integers from 1 to 7 (7x6x5x4x3x2x1) and (7-3)! means the product of all positive integers from 1 to 4 (4x3x2x1). Calculate the factorial of 7 and 4, arrange them as described by the formula and solve the problem. \(_7P_3 = \frac{7!}{4!} = \frac{7x6x5x4x3x2x1}{4x3x2x1} = 7x6x5 = 210\).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, particularly in the study of combinatorics and permutations. A factorial, denoted by an exclamation point (!), is the product of an integer and all the integers below it. For example, the factorial of 5, written as 5!, is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1
• This equals 120.

Factorials are important because they help in counting the number of ways items can be arranged. As the number increases, the factorial grows very quickly. For example, 7! = 5040, and this represents all the possible ways you can arrange 7 distinct objects. They form the basis for calculating permutations and combinations, making them a key tool in combinatorics.

Permutation Formula

Permutations are a way of arranging a set number of objects. The permutation formula itself is defined as:\[_nP_r = \frac{n!}{(n-r)!}\]Here, _nP_r denotes the number of ways to arrange 'r' items from a set of 'n' items. It's crucial to note that the order of arrangement matters when calculating permutations. This formula helps us to determine how many different sequences we can make from 'n' objects taking 'r' at a time.

For example, if you have 7 books and want to know how many ways you can arrange any 3 of them, you'd use the formula:

• _7P_3 = \(\frac{7!}{4!}\)
• As calculated: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 and 4! = 4 × 3 × 2 × 1
• Therefore, _7P_3 = 7 × 6 × 5 = 210.

This result means there are 210 different ways to select and arrange 3 books from a total of 7.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arrangement, and combination of objects. It explores how objects can be selected and arranged, often without regard to order in the case of combinations, but with order in permutations. This field is essential for analyzing situations where order or selection matters, such as arranging members in a committee or seating guests at a table.

Combinatorics encompasses a variety of principles, including:

• Permutations: Concerned with arranging items where order is important.
• Combinations: Involves selecting items where order does not matter.
• Principle of Inclusion-Exclusion: Used to calculate the number of elements in the union of multiple sets.

These principles provide the tools to solve a wide array of problems involving finite sets and discrete structures, making it highly applicable in fields such as computer science, statistics, and logistics.