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Chapter 10: Problem 1

Evaluate. $$4 !$$

Short Answer

Expert verified
The value of \(4!\) is 24.

Step by step solution

01

Understand the concept of factorial

A factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\). It is denoted by \(n!\). For instance, the factorial of 4, denoted by \(4!\), is \(4*3*2*1\)
02

Compute the factorial of 4

\(4! = 4*3*2*1 = 24.\)

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Key Concepts

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

Combinatorics
Combinatorics is a fascinating branch of mathematics that deals with counting, arranging, and combining objects. It is essential for understanding how different objects can be selected and ordered under specific rules. Factorials play a vital role in calculating the number of possible arrangements or permutations of a set. For example, if you have four people and want to know how many ways they can stand in line, you would calculate the factorial of 4, which is denoted by \(4!\). This means you multiply all numbers from 1 to 4 together to get 24 possible arrangements.
Number Theory
Number theory is the study of integers and the properties that arise from their structure and behavior. Factorials are connected to number theory because they deal with integers and their products. As you compute factorials, like \(4! = 24\), you start understanding how these products relate to properties such as divisibility and primes in a deeper sense. Analyzing factorials can unravel many secrets about how numbers interact and how they form the building blocks of more complex mathematical theories.
Multiplication
Multiplication is a fundamental operation in mathematics involving the repeated addition of a number. In the context of factorials, multiplication is key to calculating the product of sequential integers. For instance, with \(4!\), you multiply each integer from 1 to 4 (i.e., \(4 \times 3 \times 2 \times 1\)). This process not only helps in determining factorials but also builds a solid foundation for working with other mathematical concepts that require orchestrating multiple multiplicative steps.
Integers
Integers are whole numbers that can be positive, negative, or zero. They are the only type of numbers used in factorial calculations. Factorials involve multiplying several positive integers together, making them an exclusive integer operation. With factorial \(4!\), only positive integers from 1 to 4 are involved in the multiplication process. Understanding integers and their behavior under operations like multiplication is crucial when calculating and interpreting factorials in mathematical contexts or problem-solving scenarios.

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Most popular questions from this chapter

During the play of a card game, you see 20 of 52 cards in the deck drawn and discarded and none of them is a black \(4 .\) You need a black 4 to win the game. What is the probability that you will win the game on the next card drawn?

Assume that the probability of winning 5 dollars in the lottery (on one lottery ticket) for any given week is \(\frac{1}{50},\) and consider the following argument. "Henry buys a lottery ticket every week, but he hasn't won 5 dollars in any of the previous 49 weeks, so he is assured of winning 5 dollars this week." Is this a valid argument? Explain.

Consider the following experiment: pick one coin out of a bag that contains one quarter, one dime, one nickel, and one penny. What is the probability of picking a nickel?

Consider the following experiment: draw a single card from a standard deck of 52 cards. What is the probability that the card drawn is the 2 of clubs?

Induction is not the only method of proving that a statement is true. Exercises \(26-29\) suggest alternate methods for proving statements. By factoring \(n^{2}+n, n\) a natural number, show that \(n^{2}+n\) is divisible by 2
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