Problem 83 - Algebra and Trigonometry

Question

use a calculator’s factorial key to evaluate each expression. $$ \frac{20 !}{300} $$

Step-by-Step Solution

Answer

The exact answer to this calculation heavily depends on the capacity of your calculator to handle large numbers. For most practical purposes, the result will be extremely large even after division by 300 as factorials grow extremely quickly.

Step 1: Calculate Factorial

Use the factorial key on your calculator to calculate the factorial of 20. The factorial notation, \(20!\), means to multiply 20 by every positive integer less than it down to 1. On most calculators, you enter the number (in this case, 20), then press the factorial key.

Step 2: Divide by 300

After calculating the factorial, the next step is to divide the result by 300. The calculated factorial of 20 will most likely be a very large number. To ensure accuracy, enter the entire operation into the calculator at once, if possible. If not, be accurate when saving and recalling the memory value.

Step 3: State the Result

Ensure you state the result accurately. It's worth noting that the result of this operation will depend on the capacity of your calculator, and it might need to display the answer in scientific notation due to the size of the numbers involved.

Key Concepts

Calculators in Mathematics

Calculators are invaluable tools in mathematics, simplifying complex operations that would take a lot of time by hand.
They are especially handy when dealing with large numbers or intricate calculations, such as factorials. A factorial is the product of an integer and all the integers below it down to 1. For instance, calculating 20 factorial, or \(20!\), involves multiplying 20 by 19, then by 18, and so on, all the way down to 1.
This results in a very large number that is cumbersome to compute manually.

• Most scientific calculators and graphing calculators come equipped with a factorial key, designated by \(n!\).
• By simply inputting the number "20" and pressing the factorial key, the calculator quickly performs the operation for you.
• Calculators help reduce human error that might come from manual multiplication of such large sequences.

Therefore, understanding how to use a calculator's mathematical functions effectively can save time and increase accuracy.

Evaluating Expressions

In mathematics, evaluating expressions is a way of simplifying a mathematical phrase to find its value. An expression can be made up of numbers, variables, and operations.
Consider the expression \(\frac{20!}{300}\). This is an example of evaluating an expression to find its numerical value.
Breaking it down involves two main steps:

• First, calculate the factorial of 20, which is the more complex part of the expression.
• Second, divide the resulting large number by 300.

The challenge in evaluating such expressions lies in handling large numbers and ensuring precision in calculation.
By correctly following the order of operations and using tools like calculators, you can evaluate even seemingly complex expressions with confidence.

Mathematical Notation

Mathematical notation is a system of symbols that mathematicians use to write formulas and equations. It is a universal language in mathematics, helping to convey complex ideas concisely.
In our expression \(\frac{20!}{300}\), several important notations are used:

• The factorial symbol "!" indicates that you multiply a series of descending natural numbers. In this case, \(20!\) means multiplying all numbers from 20 down to 1.
• The division symbol "/" denotes that the factorial result is being divided by 300.

This notation streamlines communication, allowing mathematicians and students to understand and solve mathematical problems efficiently.
Familiarity with these symbols is crucial in mathematics, as they frequently appear in various mathematical contexts and problem-solving scenarios. Understanding notation can vastly improve your ability to grasp and resolve mathematical exercises.