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Chapter 10: Problem 3

Evaluate \(1000 ! / 998 !\) without a calculator.

Short Answer

Expert verified
Answer: The value of the expression \(\frac{1000!}{998!}\) is 999000.

Step by step solution

01

Write out the factorials

Write out the factorials \(1000!\) and \(998!\). Let's denote the final result as "res", so we will find \(res = \frac{1000!}{998!}\).
02

Use the property of factorials to simplify

Using the property that \(n! = n \times (n-1) \times \cdots \times 2 \times 1\), the numerators and denominators can be simplified as follows: \(1000! = 1000 \times 999 \times 998 \times \cdots \times 2 \times 1\) \(998! = 998 \times 997 \times \cdots \times 2 \times 1\) All the terms from \(1\) to \(998\) are common in both \(1000!\) and \(998!\), so res can be simplified as: \(res = \frac{1000 \times 999 \times 998 \times \cdots \times 2 \times 1}{998 \times 997 \times \cdots \times 2 \times 1} = \frac{1000 \times 999 \times 998!}{998!}\)
03

Cancel out the common terms

Now, we can cancel out the common terms in the numerator and the denominator: \(res = \frac{1000 \times 999 \times 998!}{998!} = 1000 \times 999\)
04

Multiply the remaining terms

Multiply the remaining terms without a calculator: \(res = 1000 \times 999 = 999000\) So, the final result is: \(res = \frac{1000!}{998!} = 999000\)

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Key Concepts

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

Simplifying Factorials
The process of simplifying factorials involves understanding how factorials expand. When you see a project like evaluating \(1000! / 998!\), it might seem overwhelming at first because of the large numbers involved. However, simplifying factorials can make the computation much more manageable.
  • Factorials are represented by the symbol \(!\), and \(n!\) means the product of all positive integers from 1 to \(n\).
  • Thus, for example, \(1000!\) represents the multiplication of all numbers from 1 to 1000.
It helps to rewrite the factorials from their expanded forms to find common terms. By identifying these, you can simplify the expression and reduce the complexity of a problem dramatically by canceling out shared factors. This is especially useful in division situations like this one, where two factorials in the fraction can lead to a much simpler calculation.
Factorial Properties
Factorials have interesting and useful properties that significantly ease computational difficulties, especially with simplification.
  • A factorial \(n!\) can be expressed as \(n \times (n-1)!\).
  • This property is particularly useful when simplifying factorials in a division problem because it allows us to break large factorials into smaller, more manageable parts.
For instance, considering \(1000! = 1000 \times 999!\), allows simplification when dividing by \(998!\). This property can be extended and applied directly to the project, resulting in a much shorter and manageable multiplication task.
Cancelling Terms
Cancelling terms is a key step when working with factorials in division. It involves eliminating common factors that appear in both the numerator and the denominator. This is possible due to the repetitive nature of factorials.
  • The terms in the denominator \(998!\) appear in the numerator \(1000!\) as part of its expansion.
  • By recognizing that these shared factors from 1 to 998 are present in both, you can cancel them out, transforming your expression and simplifying it greatly.
This technique effectively reduces the division to a more straightforward multiplication problem: \(1000 \times 999\). It is an intuitive and practical way to cut down the work involved in calculating large factorial quotients.
Multiplication Without Calculator
Calculating products like \(1000 \times 999\) without a calculator is a crucial skill. It can be handled quickly with basic arithmetic tricks.
  • Recognize that multiplying by 1000 simply shifts the decimal point three places to the right, thus, \(1000 \times 999\) can initially be visualized as \(999 \times 10^3\).
  • Think of 999 as \(1000 - 1\) and apply the distributive property: \(1000 \times 999 = 1000 \times (1000 - 1) = 1000000 - 1000\), simplifying to 999000.
Despite being a large product, little tricks like these can help perform multiplications mentally or on paper swiftly, reinforcing arithmetic skills while avoiding calculator dependency.

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