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Chapter 4: Problem 42

Compute each of \(42-50 .\) $$ \frac{4 !}{3 !} $$

Short Answer

Expert verified
The short answer based on the provided step-by-step solution is: \(42 - 50 = -8\) and \(\frac{4!}{3!} = 4\).

Step by step solution

01

Subtract 42 and 50

To find the result of the subtraction, subtract the smaller number (42) from the larger number (50): \(50 - 42 = 8\). Since 42 is smaller than 50, the result will have a negative sign: \(-8\). So, \(42 - 50 = -8\).
02

Compute the factorials

To compute the factorials in the expression \(\frac{4!}{3!}\), we first need to find the values of \(4!\) and \(3!\). - \(4! = 4 \times 3 \times 2 \times 1 = 24\) - \(3! = 3 \times 2 \times 1 = 6\)
03

Calculate the fraction

Substitute the computed factorial values into the expression: \(\frac{4!}{3!} = \frac{24}{6}\). Divide the numerator (24) by the denominator (6) to obtain the result: \(\frac{24}{6} = 4\). So, the result of the given exercise is: 1. \(42 - 50 = -8\) 2. \(\frac{4!}{3!} = 4\)

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Key Concepts

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

Factorial Calculation
The concept of factorial is fundamental in discrete mathematics, especially when dealing with permutations and combinations. A factorial of a positive integer, denoted by the exclamation mark (e.g., \(n!\)), is the product of all positive integers less than or equal to \(n\). For example, \(4!\) means taking all integers from 1 to 4 and multiplying them together: \(4 \times 3 \times 2 \times 1 = 24\). Similarly, \(3!\) is calculated as \(3 \times 2 \times 1 = 6\).

Factorials grow rapidly; even a small increase in the number can result in a significantly larger factorial. They are used in various mathematical calculations, such as finding the number of ways to arrange a set of objects.
  • Factorial of 0 is defined as 1: \(0! = 1\).
  • Factorials are only defined for non-negative integers.
  • Useful in probability, statistics, and other areas of mathematics.
Subtraction of Integers
Subtraction is one of the fundamental arithmetic operations and refers to finding the difference between numbers. For integers, subtraction can involve both positive and negative numbers. Here, you compute \(42 - 50\).

When subtracting integers, the order matters: starting with a smaller number and subtracting a larger number results in a negative value. This gives \(50 - 42 = 8\), but since the problem requires \(42 - 50\), the result is \(-8\).
  • Always start with the first number as the minuend (what is being subtracted from).
  • The second number is the subtrahend, which is subtracted from the minuend.
  • If the minuend is smaller, the result will be negative.
Fractions in Mathematics
Fractions are an essential part of mathematics, representing parts of a whole or ratios between numbers. A fraction consists of a numerator, the number above the line, and a denominator, the number below the line.

In this exercise, you find the result of the fraction \(\frac{4!}{3!}\). After calculating the factorials, you substitute the values to get \(\frac{24}{6}\). Performing the division of the numerator by the denominator gives the result: \(4\).
  • Numerator tells how many parts you have.
  • Denominator tells into how many parts the whole is divided.
  • Always simplify the fraction if possible by finding any common factors.

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Most popular questions from this chapter

\(1^{3}+2^{3}+\cdots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}\), for all integers \(n \geq 1\)

Experiment with computing values of the product \(\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right) \cdots\left(1+\frac{1}{n}\right)\) for small values of \(n\) to conjecture a formula for this product for general \(n\). Prove your conjecture by mathematical induction.

$$ 4+8+12+16+\cdots+200 $$

Compute \(4^{1}, 4^{2}, 4^{3}, 4^{4}, 4^{5}, 4^{6}, 4^{7}\), and \(4^{8}\). Make a conjecture about the units digit of \(4^{n}\) where \(n\) is a positive integer. Use strong mathematical induction to prove your conjecture.

Suppose that \(c_{0}, c_{1}, c_{2} \ldots\) is a sequence defined as follows: $$ \begin{aligned} &c_{0}=2, c_{1}=2, c_{2}=6 \\ &c_{k}=3 c_{k-3} \quad \text { for all integers } k \geq 3 \end{aligned} $$ Prove that \(c_{n}\) is even for all integers \(n \geq 0\).
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