Problem 84 - Precalculus with Limits

Question

In Exercises 79 - 86, solve for \( n \). \( _{n + 2} P_3 = 6 \cdot _{n + 2}P_1 \)

Step-by-Step Solution

Answer

The solution to the equation is \( n = 7/5 \).

Step 1: Write down the given equation

The given equation is \( _{n + 2} P_3 = 6 \cdot _{n + 2}P_1 \).

Step 2: Substitute the permutation formula

Substitute the permutation formula into the equation and get \( \frac{(n+2)!}{(n+2-3)!} = 6 \cdot \frac{(n+2)!}{(n+2-1)!} \). We then simplify it to be \( \frac{(n+2)!}{(n-1)!} = 6 \cdot \frac{(n+2)!}{(n+1)!} \).

Step 3: Simplify equation

The \( (n+2)!\) terms on the both sides of the equation cancel each other out, thus resulting in \( \frac{1}{(n-1)!} = 6 \cdot \frac{1}{(n+1)!} \). Multiply the whole equation by \((n+1)! \cdot (n-1)!\) to remove the fractions, which gives us \( (n + 1) = 6 \cdot (n - 1) \).

Step 4: Solve for n

Doing arithmetic, we simplify as follows: \( n + 1 = 6n - 6 \), which leads to \( 5n = 7 \). Solving for \( n \) gives \( n = 7/5 \).

Key Concepts

Factorial

The factorial is a mathematical concept frequently used in permutations and combinations. It is denoted by an exclamation mark "!" and signifies the product of all positive integers up to a given number. For example, the factorial of 5, written as 5!, is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are essential because they help calculate permutations and combinations by providing the number of ways to arrange or select a set number of items. In the original exercise, factorials were directly used to calculate permutations, where we replaced the permutation notation with factorial form. Manipulating these factorial expressions is key when working towards a solution in problems involving permutations.
Understanding how factorials work and their growth rate—since it increases very rapidly—is crucial for solving these kinds of algebraic problems.

Equation Solving

Equation solving involves finding the value of the variable that makes the equation true. In the provided exercise, we have an equation comprised of factorials within the permutation formulas. Solving for the variable involves several logical steps:

• Substituting factorial expressions into the given permutation equation.
• Simplifying the equation by canceling common factors.
• Isolating the variable to find its value.

In the step-by-step solution, the process began with rewriting the permutation formula using factorials. This form increases accessibility for solving, as common terms can be easily balanced or canceled on either side of the equation. The goal then becomes to meticulously simplify and rearrange the terms to solve for the desired variable, in this case, the variable is "n".
The final stages of the solution involved basic algebraic solving techniques, including rearrangement and isolating "n" on one side of the equation to deduce its value. This demonstrates how equation solving blends elements of both algebraic manipulation and logical reasoning.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to enable easier solving. It is crucial in handling mathematical problems that involve multiple steps and complex expressions like factorials and permutations.
In the original solution, one significant manipulation was simplifying the permutation expression by canceling out the factorial terms. For instance, once both sides of the equation shared \((n+2)!\), this part was canceled out, simplifying the work to be done:

• Cancel common terms on both sides.
• Clear fractions by multiplying throughout by a common denominator.
• Simplify step-by-step by ensuring only necessary expressions remain.

The problem is then further manipulated by reorganizing terms to comfortably isolate the variable, which allowed for straightforward solving. Such methods of manipulation provide a tactical approach to break down complex problems and are fundamental tools in intermediate algebra.
Proficient use of algebraic manipulation can turn intricate equations into solvable units, reducing the apparent complexity and enhancing conceptual understanding for the student.