Problem 4 - Contemporary Precalculus

Question

Evaluate the expression. $$\frac{11 !}{8 !}$$

Step-by-Step Solution

Answer

Question: Evaluate the given expression: $$\frac{11 !}{8 !}$$ Answer: 990

Step 1: Understanding Factorials

A factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Now we can use this definition to rewrite the given expression.

Step 2: Rewrite the Expression Using Factorials

Let's rewrite the expression using factorial definition to simplify it: $$\frac{11 !}{8 !} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$$

Step 3: Simplify the Expression Using Factorial Properties

Now we can cancel out terms that are common in both numerator and denominator: $$\frac{11 \times 10 \times 9 \times \cancel{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{\cancel{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}} = 11 \times 10 \times 9$$

Step 4: Calculate the Result

Lastly, we calculate the product: $$11 \times 10 \times 9 = 990$$ So the evaluated expression is: $$\frac{11 !}{8 !} = 990$$

Key Concepts

Factorial Definition

Factorials are a fundamental concept in mathematics, especially within combinatorics and algebra. Simply put,

• A factorial, denoted by the symbol "!", is the product of all positive integers up to a specified number.
• For example, the factorial of 4, written as 4!, equals \(4 \times 3 \times 2 \times 1 = 24\).
• By convention, the factorial of zero is defined as 1 (0! = 1).

Factorials grow very quickly as the number increases. For instance, 10! is 3,628,800, which is a massive number compared to 5! which is 120.
This rapid growth makes factorials both interesting objects of study and powerful tools in mathematical operations involving permutations and combinations.

Simplifying Factorial Expressions

Simplifying factorial expressions often involves canceling out repeated factors in the numerator and denominator.
This can greatly reduce complexity in a calculation and makes it easier to handle. Here's how it works:

• When dividing one factorial by another, as shown in our example \(\frac{11!}{8!}\), write out each factorial fully.
• The expression \(\frac{11!}{8!}\) becomes \(\frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}\).

By canceling the repeated terms 8 through 1, we effectively simplify the expression to \(11 \times 10 \times 9\).
Simplifying in this way not only simplifies calculations but also aids in understanding how factorial operations function at a fundamental level.

Factorial Properties

Understanding the properties of factorials can assist in simplifying and solving complex expressions.
Key properties include:

• Multiplicative Growth: The factorial of a number \(n!\) is \(n\) times the factorial of the previous number, i.e., \(n \times (n-1)!\).
• Division: Dividing factorials often simplifies by canceling out common factors, as demonstrated by \(\frac{11!}{8!}\).
• Factorial of Zero: 0! equals 1. This property is crucial for combinatorial formulas where factorials of zero occur.

These properties emphasize the utility and flexibility of factorials in mathematical computations. For instance, in permutations where the order matters, or combinations where order doesn't matter, these properties help manage large numbers and complex calculations easily.