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Chapter 10: Problem 1

Evaluate the given expression. $$ 3 ! $$

Short Answer

Expert verified
3! = 6.

Step by step solution

01

Understanding Factorials

Factorials are a mathematical operation in which you multiply a number by every positive integer below it down to one. They are denoted by an exclamation mark (!). For example, a factorial of a number n is expressed as \( n! \). The factorial of 0 is defined to be 1.
02

Applying the Factorial Definition

To find \( 3! \), we need to multiply 3 by every number below it down to 1. This means \( 3! = 3 \times 2 \times 1 \).

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Key Concepts

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

Mathematical Operations
Mathematical operations are the building blocks of mathematics, allowing us to perform calculations ranging from basic arithmetic to complex algebraic equations.
Factorials are one type of mathematical operation. They involve multiplying a series of descending natural numbers. The concept of factorial is symbolized by an exclamation mark next to a number, for instance, 3! means the factorial of 3.
The general rule for factorials is:
  • A factorial is calculated by multiplying the given number by all the positive whole numbers less than it.
  • Factorials are used in various branches of mathematics such as algebra, precalculus, and even in probability and statistics.
  • The factorial of zero, denoted by 0!, is defined to be 1 by convention.
Understanding these operations can assist in simplifying computations and solving complex equations.
Precalculus
Precalculus serves as the bridge between algebra and calculus, providing students with the foundational knowledge necessary for advanced mathematics. Factorials are introduced in precalculus because they are essential for understanding series, permutations, combinations, and probability.
In precalculus, you will explore:
  • Factorials as part of sequences and series, helping understand patterns and simplifications.
  • The role of factorials in binomial expansions using coefficients.
  • How factorials contribute to calculating permutations and combinations, which are foundational in probability.
By mastering factorials and their applications, students gain the insight needed for tackling calculus concepts efficiently.
Integer Multiplication
Integer multiplication is one of the fundamental arithmetic operations, dealing with the product of whole numbers. In the context of factorials, integer multiplication is crucial because it determines the value of factorial expressions.
For example, evaluating 3! involves integer multiplication:
  • You start with the number 3.
  • Then multiply it by every positive integer below it: 3 x 2 x 1.
  • This results in the product 6.
Integer multiplication with factorials helps to solve problems that require arranging items in order, analyzing symmetries, and breaking down complex calculations into simpler steps.

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Most popular questions from this chapter

The sequence $$ \left\\{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\right\\} $$ is known to converge (albeit very slowly) to a number \(\gamma\) called Euler's constant. Calculate at least the first 10 terms of the sequence. Conjecture the limit of the sequence.

Determine whether the given sequence converges. $$ \left\\{4+\frac{3^{n}}{2^{n}}\right\\} $$

Determine whether the given infinite geometric series converges. If convergent, find its sum. $$ \sum_{k=1}^{\infty} \pi^{k}\left(\frac{1}{3}\right)^{k-1} $$

Evaluate \(C(n, r)\). $$ C(50,2) $$

Determine whether the given sequence converges. $$ \left\\{\frac{10 e^{n}-3 e^{-n}}{2 e^{n}+e^{-n}}\right\\} $$
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