In Exercises 81–85, use a calculator's factorial key to evaluate each expression.
$the fraction with numerator 200 factorial and denominator 198 factorial$

Verified step by step guidance

Recall the definition of factorial: for a positive integer n, n! = n × (n-1) × (n-2) × ... × 1.

Express the given expression 200!/198! in terms of factorials to see if it can be simplified before calculating.

Write 200! as 200 × 199 × 198! to factor out 198! from the numerator and denominator.

Cancel out the common factorial term 198! in the numerator and denominator, leaving you with 200 × 199.

Use the calculator to multiply 200 by 199 to find the value of the expression 200!/198!.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factorial Definition

The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow very quickly as n increases.

Simplifying Factorial Expressions

When dividing factorials like 200!/198!, common terms cancel out. Since 200! = 200 × 199 × 198!, the expression simplifies to 200 × 199. This simplification avoids calculating large factorials directly.

Using a Calculator's Factorial Function

Many calculators have a factorial key (!) to compute factorials of integers. For large numbers, direct factorial calculation may be impractical, so simplifying expressions before using the calculator is essential to avoid errors or overflow.

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Textbook Question

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A club with 15 members is to choose four officers–president, vice president, secretary, and treasurer. In how many ways can these offices be filled?

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