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Chapter 12: Problem 1

Evaluate the expression. $$6 !$$

Short Answer

Expert verified
Answer: The value of the expression "6!" is 720.

Step by step solution

01

Understand the Factorial Concept

A factorial of a non-negative integer n, which is denoted by n! is the product of all positive integers less than or equal to n. In this case, we have to find 6! which can be written as 6 脳 5 脳 4 脳 3 脳 2 脳 1.
02

Multiply the Numbers

To find the value of 6!, we multiply all the numbers from 1 to 6 as follows: $$6! = 6 脳 5 脳 4 脳 3 脳 2 脳 1$$
03

Calculate the Result

Let's perform the multiplication: $$6! = 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 720$$ 6! is equal to 720.

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Key Concepts

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

Mathematics
In mathematics, the concept of factorial is a fundamental operation, particularly when dealing with permutations and combinations. A factorial, denoted by the symbol "!", represents the product of all positive integers up to a given number. For example, the factorial of 6, which is written as 6!, means multiplying 6 by all the integers less than it down to one, i.e., 6 \( \times \) 5 \( \times \) 4 \( \times \) 3 \( \times \) 2 \( \times \) 1. This equals to 720.
Understanding factorials is crucial for various topics in mathematics, as they lay the groundwork for more complex operations like binomial theorem expansions and limits involving series.
Combinatorics
Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. Factorials play a central role here because they help calculate the number of ways objects can be arranged or ordered.
For instance, when determining the number of ways to arrange 6 distinct items, the answer is 6!, which is 720.
Factorials contribute to two main combinatorial ideas:
  • Permutations: The arrangement of items in specific order, where order matters. For arranging all items, the factorial of the item count is used.
  • Combinations: Selecting items where the order does not matter, often using factorials in combination with other functions.
By using these factorial-based operations, combinatorics offers insightful methods to solve various real-world problems from coding to cryptography.
Arithmetic Operations
Arithmetic operations include basic mathematical operations: addition, subtraction, multiplication, and division. Factorials specifically involve multiplication, as they require multiplying a series of consecutive natural numbers up to the specified number.
For example, calculating 6! in arithmetic involves repeated multiplication steps:
  • First, multiply the largest two numbers: 6 \( \times \) 5 = 30
  • Next, multiply 30 by 4: 30 \( \times \) 4 = 120
  • Continue with 120 \( \times \) 3 = 360
  • Multiply by 2: 360 \( \times \) 2 = 720
  • Finally, multiply by 1, remaining at 720
This process underlines the foundational nature of multiplication within the realm of factorials in arithmetic operations. Understanding how to execute these steps is crucial for honing arithmetic skills and solving factorial-related problems efficiently.

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Most popular questions from this chapter

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