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Chapter 9: Problem 75

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

Short Answer

Expert verified
After going through the process, the solution to this problem is \( n = 2 \).

Step by step solution

01

Understand what the problem is asking

The problem is asking to solve for the variable \( n \) in the equation \( 4 \cdot _{n+1} P_{2} = _{n+2} P_{3} \).
02

Use the permutation formula

Substitute the permutation formula into the equation. Therefore, the equation becomes \( 4 \cdot (n+1)! / (n+1-2)! = (n+2)! / (n+2-3)!.
03

Simplify the equation

By simplifying the equation, we rewrite it as \( 4 \cdot (n + 1)n = (n+2)(n+1)n \).
04

Solve for the variable

To solve for \( n \), we can then divide through by \( (n+1)n \), and next subtract \( 4 \) from both sides to isolate \( n \). In doing this, we find that \( n + 2 = 4 \), and consequently, \( n = 2 \).

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Key Concepts

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

Permutation Formula
Permutations are fundamental to understanding arrangements in mathematics, especially where order matters. The permutation formula is given by \( _nP_r = \frac{n!}{(n-r)!} \), where 'n' represents the total number of objects and 'r' signifies the number selected. Factorial notation (!) is used in this context, which can also be a source of confusion.

In our problem, the permutation formula is applied to find the number of ways to arrange 'r' objects out of 'n'. With \( _{n+1} P_{2} \) and \( _{n+2} P_{3} \), the formula helps translate the problem into algebraic terms involving factorials, which can then be manipulated to solve for 'n'. Understanding this formula is key as it serves as the foundation for solving many types of combinatorial problems.
Factorial Notation
Understanding factorial notation is crucial for solving permutation problems. A factorial, represented by an exclamation point (!), is the product of all positive integers up to a certain number. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

It's essential to note that \(0! = 1\), a definition which may seem counterintuitive, but is important for the mathematical properties of the factorial function. In our given problem, the use of factorial notation simplifies the process of handling the permutations. By using the factorial, expressions that initially seem complex become more manageable, making it easier to carry out algebraic manipulation to find the desired variable 'n'.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying an equation to make it solvable鈥攁n essential skill in mathematics. It typically involves operations such as adding, subtracting, dividing, multiplying, and factoring. For permutation equations, this often means breaking down factorial expressions and canceling out common terms.

The given exercise involves multiple steps of algebraic manipulation: dividing through by common factors (to cancel out terms), isolating the variable (finding 'n' on one side of the equation), and simplifying the equation to the most straightforward form to solve for 'n'. Applied properly, these techniques turn complex-looking permutation equations into soluble algebraic ones, paving the way to arrive at the solution efficiently.

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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. $$\sum_{n=1}^{10} n^{3}$$

Simplifying a Difference Quotient In Exercises \(67-72\) , simplify the difference quotient, using the Binomial Theorem if necessary. $$\frac{f(x+h)-f(x)}{b} \quad$$ Difference quotient $$f(x)=\frac{1}{x}$$

Expanding an Expression In Exercises \(61-66,\) use the Binomial Theorem to expand and simplify the expression. $$\left(u^{3 / 5}+2\right)^{5}$$

Graphical Reasoning In Exercises 83 and \(84,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. $$f(x)=-x^{4}+4 x^{2}-1, \quad g(x)=f(x-3)$$

Determine whether the statement is true or false. Justify your answer. If \(A\) and \(B\) are independent events with nonzero probabilities, then \(A\) can occur when \(B\) occurs.
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