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Chapter 9: Problem 77

In Exercises \(75-82,\) solve for \(n\) $$_{n+1} P_{3}=4 \cdot_{n} P_{2}$$

Short Answer

Expert verified
The solution to the equation \(_{n+1} P_{3}=4 \cdot_{n} P_{2}\) is \(n = 3\).

Step by step solution

01

Convert Permutation Notation to Factorials

We first convert the permutation equation, \(_{n+1} P_{3}=4 \cdot_{n} P_{2}\), to factorial notation. This gives us the equation \((n+1)! / ((n+1)-3)! = 4 * (n! / (n-2)!)\). Which simplifies to \((n+1)! / (n-2)! = 4 * (n! / (n-2)!)\).
02

Clear the Fractions and Simplify

To clear the fractions, we can multiply both sides by \((n-2)!\), which yields \((n+1)! = 4 * n!\). Now simplify the equation by expanding factorials. This yields \((n+1)*n! = 4 * n!\).
03

Divide by Common Terms and Solve for n

We can then further simplifiy by dividing both sides by \(n!\). This gives \((n+1) = 4\). Solving for \(n\) we will get \(n = 4 - 1\), thus \(n = 3\).

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Key Concepts

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

Factorials
Factorials are an essential concept in mathematics, particularly in permutations and combinations. A factorial of a number, denoted as !, means multiplying the number by every positive integer less than itself. For example, the factorial of 4, written as 4!, is calculated as:
  • 4! = 4 脳 3 脳 2 脳 1 = 24
Factorials help us determine the number of possible arrangements or sequences of a set of elements. These are used extensively in permutation problems where order matters. Factorial notation simplifies equations involving large numbers by replacing sequences of multiplications. In the given exercise, translating permutation notation to factorials allows the problem to be expressed in a simpler mathematical form. Understanding this conversion is crucial to solving permutation equations effectively.
Solving Equations
Solving equations involving permutations often requires a step-by-step approach. The goal is to isolate the variable, in this case, \(n\). Starting with the equation \[\frac{(n+1)!}{(n+1-3)!} = 4 \times \frac{n!}{(n-2)!}\] we look to remove fractions to make calculations more straightforward. The process involves equating and expanding factorials while simplifying expressions on both sides. Key steps include:
  • Clearing fractions by multiplying both sides by denominators
  • Simplifying expressions, using algebraic manipulation such as expanding and canceling factorials
  • Dividing by common terms to reveal the variable
In our exercise, we simplify and solve, finding that \(n\) equals 3. Breaking complex equations into simpler parts aids in understanding and accurately determining solutions.
Simplification Techniques
Simplification techniques are vital in breaking down complicated mathematical expressions into more manageable forms. In our context, converting a permutation equation into its factorial notation is a form of simplification that allows easier manipulation and solving of the equation. After translation, the simplification process includes:
  • Removing fractions by multiplying through, making calculations simpler
  • Expanding and canceling factorial terms to reduce the equation
  • Employing algebraic divisions to eliminate common factors
These steps are strategic and lead to a clearer path for finding solutions. By simplifying expressions, mathematicians can focus on the core elements of a problem, thereby isolating the variable of interest, like \(n\) in our exercise. Understanding and applying these techniques makes solving complex problems both efficient and less daunting.

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Most popular questions from this chapter

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{n=1}^{30} n$$

Linear Model, Quadratic Model, or Neither? In Exercises \(61-68\) , write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. $$a_{2}=-3$$ $$a_{n}=-2 a_{n-1}$$

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{n=1}^{20}\left(n^{3}-n\right)$$

Finding a Sum In Exercises \(45-54\) , find the sum using the formulas for the sums of powers of integers. $$\sum_{j=1}^{10}\left(3-\frac{1}{2} j+\frac{1}{2} j^{2}\right)$$

Approximation In Exercises \(79-82,\) use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise \(79,\) use the expansion \((1.02)^{8}=(1+0.02)^{8}\) $$=1+8(0.02)+28(0.02)^{2}+\cdots+(0.02)^{8}$$ $$(1.98)^{9}$$
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