Problem 48 - Algebra for College Students

Question

The quantity \(P(8,3)\) can be expressed completely in factorial notation as follows: $$ P(8,3)=\frac{P(8,3) \cdot 5 !}{5 !}=\frac{(8 \cdot 7 \cdot 6)(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}{5 !}=\frac{8 !}{5 !} $$ Express each of the following in terms of factorial notation. (a) \(P(7,3)\) See below (b) \(P(9,2)\) See below (c) \(P(10,7)\) See below (d) \(P(n, r), \quad r \leq n\) and 0 ! is defined to be \(1 \frac{n !}{(n-r) !}\)

Step-by-Step Solution

Answer

(a) \(P(7,3) = \frac{7!}{4!}\), (b) \(P(9,2) = \frac{9!}{7!}\), (c) \(P(10,7) = \frac{10!}{3!}\), (d) \(P(n,r) = \frac{n!}{(n-r)!}\).

Step 1: Understanding Permutations

The number of permutations of choosing \( r \) objects from \( n \) objects, denoted as \( P(n, r) \), is calculated as \( \frac{n!}{(n-r)!} \). This formula accounts for the arrangements of the chosen \( r \) objects in a sequence.

Step 2: Evaluate (a) \(P(7,3)\)

Using the permutation formula, express \( P(7, 3) \) as:\[ P(7, 3) = \frac{7!}{(7-3)!} = \frac{7!}{4!} \]

Step 3: Evaluate (b) \(P(9,2)\)

Using the formula for permutations, express \( P(9, 2) \) as:\[ P(9, 2) = \frac{9!}{(9-2)!} = \frac{9!}{7!} \]

Step 4: Evaluate (c) \(P(10,7)\)

Using the formula for permutations, express \( P(10, 7) \) as:\[ P(10, 7) = \frac{10!}{(10-7)!} = \frac{10!}{3!} \]

Step 5: Evaluate (d) General Case \(P(n, r)\)

For the general case when \( r \leq n \), the permutation expression is: \[ P(n, r) = \frac{n!}{(n-r)!} \]

Key Concepts

Factorial Notation

Factorial notation is a mathematical concept used to express the product of a sequence of descending positive integers. For example, when we use the notation "8!", it means that we are multiplying the numbers from 8 down to 1: \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]This results in the product of all these numbers. Factorials are very useful in various fields of mathematics, especially in permutations and combinations. When performing calculations involving permutations, factorial notation allows us to simplify expressions and denote large product calculations concisely.Additionally, it's standard in mathematics to define \(0!\) as 1. This might seem a bit bizarre at first, but it's necessary for the mathematics of combinations and permutations to work as expected.

Permutation Formula

The permutation formula is key to solving problems where the arrangement of items is crucial. In permutations, not only is the selection of items important, but also their order.The formula for permutations is expressed as:\[ P(n, r) = \frac{n!}{(n-r)!} \]Where: - \( n \) is the total number of items,- \( r \) is the number of items to be arranged, and- \( n! \) is the factorial of \( n \), and \((n-r)!\) is the factorial of the difference between \( n \) and \( r \).This formula calculates the number of ways \( r \) items can be arranged out of \( n \) available items. By using factorial notation, the permutation formula efficiently manages large calculations. For example, to express \( P(7, 3) \), you would write:\[ \frac{7!}{(7-3)!} = \frac{7!}{4!} \] This kind of formula is fundamental to combinatorial mathematics and is often used in problems involving sequence and arrangement.

Combinatorial Mathematics

Combinatorial mathematics is a vast field that studies the ways in which discrete objects can be arranged, selected, and grouped. Permutations are a core concept in this branch of mathematics.Permutations are concerned with the arrangement of a set number of objects, where order matters. In combinatorial terms, if you have a set of \( n \) items and wish to select \( r \) of them, permutations allow us to determine how many different ways these items can be ordered.This is different from combinations, where the order does not matter. In permutations, each different order counts as a unique arrangement. For instance, arranging \( A, B, \text{and} C \) is different from arranging \( B, A, \text{and} C \). Formulas like the permutation formula:\[ P(n, r) = \frac{n!}{(n-r)!} \] help in computing the number of possible arrangements, making it a cornerstone tool in solving combinatorial problems. Combinatorial mathematics is heavily used in computer science, genetics, and many other fields where complex arrangement and selection problems need solving.