Problem 42
Question
Solve each equation. $$P(n, 2)=42$$
Step-by-Step Solution
Verified Answer
The solution to the given equation is \(n=5\).
1Step 1: Write the Permutations formula for r = 2
We will first write the formula for permutations with r = 2:
$$
P(n, 2) = \frac{n!}{(n-2)!}
$$
2Step 2: Substitute 42 into the formula
Plug 42 into the formula, resulting in:
$$
42 = \frac{n!}{(n-2)!}
$$
3Step 3: Multiply by (n-2)! on both sides of the equation
Multiply both sides of the equation by (n-2)! to eliminate the denominator:
$$
42(n-2)! = n!
$$
4Step 4: Find the value of n that satisfies the equation
Start by testing integers for n, and calculate both sides of the equation.
For example, let's start with n = 3. If n = 3, then:
Left side of the equation:
$$
42(3-2)! = 42(1)! = 42
$$
Right side of the equation:
$$
3! = 6
$$
Since the left side is not equal to the right side, n = 3 does not satisfy the equation.
Next, try n = 4:
Left side of the equation:
$$
42(4-2)! = 42(2)! = 168
$$
Right side of the equation:
$$
4! = 24
$$
Again, the left side is not equal to the right side, so n = 4 does not satisfy the equation.
Finally, try n = 5:
Left side of the equation:
$$
42(5-2)! = 42(3)! = 252
$$
Right side of the equation:
$$
5! = 120
$$
For n = 5, the left side is equal to the right side, and so when n = 5, the equation is satisfied.
5Step 5: Write the solution
The solution to the given equation is:
$$
n = 5
$$
Key Concepts
Factorial NotationSolving Permutations EquationsCombinatorial Analysis
Factorial Notation
Understanding factorial notation is essential when dealing with permutations in discrete mathematics. A factorial, represented by an exclamation point (!), refers to the product of all positive integers less than or equal to a given number. For instance, the factorial of 5, denoted as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials are the building blocks of permutations because they represent the number of ways to arrange a certain number of objects. When you see an equation involving permutations, like \(P(n, 2) = 42\), it relies heavily on understanding factorials to manipulate and solve it.
While dealing with factorial notation, remember that \(0!\) is defined to be 1. This is because the number of ways to arrange zero objects is just one way—which is to arrange nothing.
Factorials are the building blocks of permutations because they represent the number of ways to arrange a certain number of objects. When you see an equation involving permutations, like \(P(n, 2) = 42\), it relies heavily on understanding factorials to manipulate and solve it.
While dealing with factorial notation, remember that \(0!\) is defined to be 1. This is because the number of ways to arrange zero objects is just one way—which is to arrange nothing.
Solving Permutations Equations
To solve permutations equations, you must first be acquainted with the permutations formula, which is \(P(n, r) = \frac{n!}{(n-r)!}\), where \(n\) represents the total number of items and \(r\) represents the number of items to arrange.
For the given problem, \(P(n, 2) = 42\), you substitute the known values and solve for \(n\). The steps involve simplifying the factorial expressions and methodically increasing the value of \(n\) until you find an integer that satisfies both sides of the equation. This process shows the practical application of both permutation concepts and the factorial notation in solving equations.
Moreover, understanding of 'solving permutations equations' can be deepened by repeatedly performing similar exercises, which will enhance the ability to solve more complex problems involving permutations.
For the given problem, \(P(n, 2) = 42\), you substitute the known values and solve for \(n\). The steps involve simplifying the factorial expressions and methodically increasing the value of \(n\) until you find an integer that satisfies both sides of the equation. This process shows the practical application of both permutation concepts and the factorial notation in solving equations.
Moreover, understanding of 'solving permutations equations' can be deepened by repeatedly performing similar exercises, which will enhance the ability to solve more complex problems involving permutations.
Combinatorial Analysis
Combinatorial analysis is the study of counting, arrangement, and combination of objects. Within combinatorial analysis, permutations are just one type of counting problem. They deal with the number of ways of ordering distinct objects.
In the problem \(P(n, 2) = 42\), the aim is to find how many distinct ways two objects can be arranged from a larger set. This is a classic example of using combinatorial analysis to solve real-world problems. In such problems, factorial notation plays a key role, as the concepts are deeply interlinked.
Understanding combinatorial analysis allows students to approach problems logically and systematically, crucial skills in mathematics and various fields where discrete mathematics is applied, such as computer science, cryptography, and operations research. By mastering combinatorial analysis, students can tackle a wide array of problems involving counting principles.
In the problem \(P(n, 2) = 42\), the aim is to find how many distinct ways two objects can be arranged from a larger set. This is a classic example of using combinatorial analysis to solve real-world problems. In such problems, factorial notation plays a key role, as the concepts are deeply interlinked.
Understanding combinatorial analysis allows students to approach problems logically and systematically, crucial skills in mathematics and various fields where discrete mathematics is applied, such as computer science, cryptography, and operations research. By mastering combinatorial analysis, students can tackle a wide array of problems involving counting principles.
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