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Chapter 10: Problem 23

Use factorial notation to rewrite the given product. $$ 4 \cdot 3 $$

Short Answer

Expert verified
The product \(4 \cdot 3\) is \(\frac{4!}{2!}\).

Step by step solution

01

Identify the Product

The given product is written as \(4 \cdot 3\). It is the multiplication of two consecutive integers: 4 and 3.
02

Express as Factorials

We know that the factorial of a number \(n\), denoted as \(n!\), is the product of all positive integers up to \(n\). Therefore, the factorial of 4 written as 4! is: \[ 4! = 4 \cdot 3 \cdot 2 \cdot 1 \].
03

Simplify the Factorial Expression

Since \(4! = 4 \cdot 3 \cdot 2 \cdot 1\), we can express \(4 \cdot 3\) as \(\frac{4!}{2!}\) because\[ \frac{4!}{2!} = \frac{4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1} = 4 \cdot 3 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Consecutive Integers
Consecutive integers are numbers that follow each other in order without any gaps. For example, 1, 2, 3, and 4 are consecutive integers, as are 4 and 3. Each number differs from the preceding one by exactly one. In mathematical problems and especially those involving multiplication, identifying consecutive integers is important. This is because multiplications involving consecutive integers can sometimes be expressed in a simpler or related form, such as through factorial notation.
  • Example: In the product \(4 \cdot 3\), the numbers 4 and 3 are consecutive integers.
  • Consecutive integers help simplify expressions by linking them to factorial notation for easier computation.
Mathematical Factorials
Factorials are a fundamental concept in mathematics, often used in permutations, combinations, and various algebraic expressions. The factorial of a number, represented as \(n!\), is the product of all positive integers up to that number. For instance, the factorial of 4, noted as \(4!\), is calculated as \(4 \cdot 3 \cdot 2 \cdot 1\).
  • Factorials grow very quickly; for example, \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\).
  • Factorial notation simplifies complex multiplications by providing a shorthand to express repetitive products.
When rewriting a multiplication problem using factorials, you can break down or build up the product using known factorials. This process can reveal simplified forms of expressions.
Simplifying Expressions
Simplifying mathematical expressions often involves rewriting them in a form that is easier to understand or compute. In problems involving factorials, we simplify by canceling out terms or factors in the numerator and the denominator. Consider the example of simplifying \(4 \cdot 3\) using factorials. We start with the known factorial \(4!\), which is \(4 \cdot 3 \cdot 2 \cdot 1\).
  • To simplify \(4 \cdot 3\), recognize that it can be expressed as \(\frac{4!}{2!}\).
  • Here, \(2! = 2 \cdot 1\), so dividing \(4!\) by \(2!\) cancels the \(2 \cdot 1\) terms out, leaving \(4 \cdot 3\).
This step not only simplifies the expression but also provides a deeper understanding of how factorials can be manipulated to achieve simpler forms of what initially appear as complicated products.

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