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Chapter 11: Problem 8

Evaluate. $$1 !$$

Short Answer

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Step by step solution

01

Understand the factorial notation

The exclamation mark (!) represents the factorial of a number. The factorial of a number is the product of all positive integers less than or equal to that number.
02

Apply the factorial formula

The formula for the factorial of a number n is given by \[ n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \].
03

Calculate 1!

Using the factorial formula, substitute 1 into it: \[ 1! = 1 \].

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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 exclamation mark (!) to represent the multiplication of a series of descending natural numbers. For example, in the notation 4!, the exclamation mark means you multiply all positive integers from 4 down to 1. This notation is especially useful in combinatorics and probability, where calculations often involve arranging or selecting items.
To summarize: when you see a number followed by an exclamation mark, it's calling for the factorial function, which means to multiply that number by every positive integer less than itself.
Factorial Formula
The factorial formula expresses the product of all positive integers up to a given number, n. Mathematically, it鈥檚 written as:
\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 1\].
The formula tells you to start with the number n and keep multiplying by the next smaller number, until you reach 1.
Let's look at some examples to understand how the formula works:
  • For 3!, you calculate \[ 3\times 2 \times 1 = 6 \].
  • For 5!, the calculation goes \[ 5 \times 4 \times 3 \times 2 \times 1 = 120 \].
Notice how each number reduces by 1 in each step, continuing until you reach 1.
Basic Arithmetic
Basic arithmetic refers to simple mathematical operations including addition, subtraction, multiplication, and division. Understanding these operations is crucial when working with factorials, as they often require you to perform multiple-step multiplications.
For example, calculating 4! involves successive multiplications:
  • First, 4 times 3, which equals 12,
  • Second, multiply that 12 by 2 to get 24,
  • Finally, multiply 24 by 1 to get 24.
If you know how to multiply and follow the steps sequentially, you can solve factorial problems easily.
Therefore, mastering the basic operations helps you perform more complex tasks like evaluating factorials.
Master these core concepts to handle factorials with ease!

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