Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the factorial. $$ \frac{100 !}{99 !} $$

Step-by-Step Solution

Answer

The value is 100.

Step 1: Understand the Factorial Notation

Factorial of a number, denoted as \(!\), is the product of all positive integers less than or equal to that number. For example, \(n! = n \times (n-1) \times (n-2) \times \, ... \, \times 1\). When working with \(n!\) where \(n\) is a positive integer, it essentially means multiplying the number by every integer below it down to 1.

Step 2: Simplify the Expression

Given the expression \(\frac{100!}{99!}\), recognize that both the numerator \(100!\) and the denominator \(99!\) have similar factorial terms. The expression can be simplified by noting that \(100! = 100 \times 99!\).

Step 3: Cancel Out Common Terms

Substitute \(100!\) in the original expression: \(\frac{100 \times 99!}{99!}\). Notice that \(99!\) appears in both the numerator and the denominator, so they can be canceled out. This simplifies the expression to \(100\).

Key Concepts

Factorial Notation

Factorial notation is a mathematical concept that helps solve problems involving permutations and combinations. It is denoted by an exclamation mark "!" placed after a number. A factorial of a number, say \(n\), represented as \(n!\), is the product of all positive integers from 1 to \(n\). This means \(n! = n \times (n-1) \times (n-2) \times \, ... \, \times 1\). Factorials rapidly increase as the number becomes larger.
For example:

• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Factorial notation is especially helpful when calculating probabilities, arranging items, or understanding sequences in mathematics. This notation requires understanding how to expand a number into its factorial form, ensuring accurate evaluations and simplifications later in mathematical equations.

Simplifying Expressions

Simplifying expressions in mathematics often involves using the properties of numbers or operations to make them easier to handle. When dealing with factorials, this means recognizing that parts of the factorial terms can cancel out, particularly in fractions. Take the example given in the original exercise: \(\frac{100!}{99!}\).
Given \(100! = 100 \times 99 \times 98 \times \, ... \, \times 1\) and \(99! = 99 \times 98 \times \, ... \, \times 1\), it becomes clear that their similarity can be used to simplify the expression. Since \(100!\) can be rewritten as \(100 \times 99!\), substitution can be performed:

• Numerator becomes \(100 \times 99!\)
• Denominator remains \(99!\)

This results in a cancelation of \(99!\) in both the numerator and the denominator, simplifying the expression to just \(100\). Such simplifications make calculations more manageable and less time-consuming, illustrating the importance of understanding mathematical structures and relationships.

Factorial Properties

Factorial properties are mathematical rules that help manipulate expressions involving factorials, leading to easier computation and understanding. A fundamental property is that for any positive integer \(n\), the factorial \(n!\) can be expressed in terms of the factorial of the previous integer:
\(n! = n \times (n-1)!\).
This is the key property used in the exercise to simplify the expression. By recognizing that the sequence of multiplication within \(n!\) is filled with integers leading down to 1, it is possible to deduce shortcuts in calculations.

• For example, \(5! = 5 \times 4!\), since \(5 \times 4 \times 3 \times 2 \times 1\) and \(4 \times 3 \times 2 \times 1\) allow part of the sequence to be factored out or expanded.
• This can be particularly powerful in simplifying complex expressions where factorials are in both the numerator and the denominator.

Factorial properties provide efficiency and consistency, making them an essential tool in combinatorics and other branches of mathematics where arrangements and repetitions are significant aspects of calculation.