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Chapter 15: Problem 14

Evaluate. $$6 !$$

Short Answer

Expert verified
To evaluate \(6!\), we expand it as \(6 * 5 * 4 * 3 * 2 * 1\) and then multiply all the numbers together to get \(6! = 720\).

Step by step solution

01

Write down the factorial expression

For the given problem, we need to find the value of \(6!\). So, expand it as follows: \(6! = 6 * 5 * 4 * 3 * 2 * 1\)
02

Multiply the numbers

Now, multiply all the numbers in the expression: \(6! = 6 * 5 * 4 * 3 * 2 * 1 = 720\)
03

Write the final answer

We found the value of \(6!\), so the final answer is: \(6! = 720\)

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Key Concepts

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

Permutations
Permutations are arrangements or sequences of items where the order is significant. Imagine you have a stack of books, and you want to arrange three of them on a shelf. If you change the order in which they are set, you have different permutations.
When calculating permutations, factorials play a central role. For instance, if you want to know how many different ways you can arrange 6 distinct objects, you will use the factorial:
  • The operation for arranging 6 distinct items is represented as \(6!\).
  • The formula in terms of permutations is \(P(n) = n!\), where \(n\) is the number of items.
  • For 6 items, \(6! = 720\) means there are 720 different ways to arrange the 6 items in a sequence.
Understanding permutations is fundamental in determining probabilities where order matters, such as seating arrangements, ranking, or sequences.
Combinatorics
Combinatorics is a branch of mathematics focused on counting, either as different permutations or combinations. It helps solve problems related to how objects can be selected and arranged. Two major problems tackled by combinatorics are:
  • **Permutations** - Arrangements where order is important. Already explained, using factorials to determine the number of sequences.
  • **Combinations** - Selections where order does not matter. When choosing items from a set without considering the sequence, such as selecting 3 fruits out of 6, combinations are used.
In permutations, factors like factorial calculations become primary tools. In our original exercise, evaluating \(6!\) is vital because it forms the basis of combinatorial problems dealing with arrangements. Combinatorics helps in simplifying complex arrangement and selection problems, such as pathways, allocation, and resource distribution.
Number Theory
Number theory deals with integers and their fascinating properties. It is considered the purest study of number systems and focuses on understanding the nature and relationships among numbers.
One aspect that aligns with our exercise on factorials is the importance of understanding divisibility and prime factors, especially when working with large numbers like factorials.
  • **Divisibility** - Factorials are divisible by all integers smaller than or equal to the number being factorialized. So, \(6!\) is divisible by 1, 2, 3, 4, 5, and 6.
  • **Prime Factorization** - Involves expressing numbers as a multiplication of prime numbers. For example, \(720\) can be broken down into primes as \(2^4 \times 3^2 \times 5^1\).
These properties can be highly useful in solving complex number theory problems, such as finding common multiples or solving Diophantine equations. In essence, factorials connect various concepts within number theory, demonstrating the deep interrelations between different mathematical areas.

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Most popular questions from this chapter

Find the sum of the terms of the infinite geometric sequence, if possible. $$36,6,1, \frac{1}{6}, \dots$$

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Use Pascal鈥檚 Triangle to expand each binomial. $$(k+2)^{5}$$

Use the binomial theorem to expand each expression. $$(2 k+1)^{4}$$

Find the sum of the terms of the infinite geometric sequence, if possible. $$8, \frac{16}{3}, \frac{32}{9}, \frac{64}{27}, \dots$$
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