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Chapter 14: Problem 16

Evaluate each expression. $$ \frac{5 !}{5 ! 0 !} $$

Short Answer

Expert verified
The value of the expression is 1.

Step by step solution

01

Understand the Factorial Notation

Factorials are denoted by the symbol '!', and they represent the product of an integer and all positive integers less than it. For instance, 5! = 5 脳 4 脳 3 脳 2 脳 1.
02

Calculate 5!

Evaluate 5!: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
03

Calculate 0!

Recall that by definition, 0! is equal to 1. Hence, \( 0! = 1 \).
04

Simplify the Expression

Substitute the values of 5! and 0! into the expression: \( \frac{5!}{5! \times 0!} = \frac{120}{120 \times 1} \)
05

Calculate the Final Value

Simplify the fraction: \( \frac{120}{120} = 1 \).

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Key Concepts

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

Factorial Notation
Factorial notation uses the symbol '!' to represent the product of an integer and all positive integers less than it. This might sound complex at first, but it gets simpler with examples. For instance, if you see 5!, it stands for 5 multiplied by every whole number less than 5 down to 1. So, 5! is calculated as follows:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
Notably, there's a special case with zero: 0!. By definition, 0! is equal to 1. Even though it might seem counterintuitive, it is essential in various mathematical applications.
To summarize factorial notation:
  • Factorial of any positive integer n is n! = n 脳 (n-1) 脳 ... 脳 1
  • 0! is always 1
Simplifying Expressions
When faced with an expression involving factorials, it often looks more complex than it is. Simplifying such expressions step-by-step can clear things up.
Let's break down the original expression:
\[ \frac{5!}{5! \times 0!} \]By substituting the values:
\[ \frac{120}{120 \times 1} \]
This fraction can now be simplified. 120 divided by 120 equals 1:
\[ \frac{120}{120} = 1 \]
Breaking expressions down like this helps make them manageable and allows you to see how each part of the expression contributes to the final result.
Division of Factorials
Dividing factorials might look tricky, but it follows similar rules to dividing other numbers. Consider our original problem again:
\[ \frac{5!}{5! \times 0!} \]
Here, we have factorials both in the numerator (top) and the denominator (bottom) of our fraction. The key steps involve:
  • Calculating each factorial separately 鈥� as we've seen, 5! equals 120, and 0! equals 1.
  • Substituting these values back into the fraction.
  • Simplifying the resulting fraction through division.
In this case, the expression simplifies naturally:
\[ \frac{120}{120 \times 1} = \frac{120}{120} = 1 \]
Understanding these steps makes it easier to handle more complex expressions involving the division of factorials in future problems.

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Most popular questions from this chapter

Write the first five terms of each arithmetic sequence. \(a_{1}=-2, d=-4\)

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \ldots $$

Determine an expression for the general term \(a_{n}\) of each sequence $$ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots $$

Fill in each blank with the correct response. The value of the sum \(\sum_{i=1}^{3}(i+2)\) is ____.

Determine an expression for the general term of each arithmetic sequence. Then find \(a_{25}\). \(-3,0,3, \ldots\)
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