Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ P(n, n-2) $$

Step-by-Step Solution

Answer

The short answer for the given expression is: \( P(n, n-2) = \frac{n \cdot (n-1) \cdot (n-2)!}{2} \).

Step 1: Write down the given expression

We are given the expression: \(P(n, n-2)\)

Step 2: Use the Permutation formula

Now, let's use the formula for permutation, which is \(P(n, r) = \frac{n!}{(n-r)!}\). In our case, n remains n, and r becomes n-2. So our formula becomes: \(P(n, n-2) = \frac{n!}{(n-(n-2))!}\)

Step 3: Simplify the expression further

Next, we need to simplify the expression inside the factorial in the denominator: \(n-(n-2) = n-n+2 = 2\) Now the expression becomes: \(P(n, n-2) = \frac{n!}{2!}\)

Step 4: Compute the factorials

To finish the evaluation, we need to compute the factorials of n and 2: \(n! = n \cdot (n-1) \cdot (n-2)! \) \(2! = 2 \cdot 1 = 2\)

Step 5: Write the final expression

Finally, we can plug the factorials back into the expression, and we get: \(P(n, n-2) = \frac{n!}{2!} = \frac{n \cdot (n-1) \cdot (n-2)!}{2}\) So the final expression for the given problem is: \( P(n, n-2) = \frac{n \cdot (n-1) \cdot (n-2)!}{2} \)

Key Concepts

Factorial

Factorials play a crucial role in permutations and combinations. A factorial, denoted as \( n! \), is the product of all positive integers from \( 1 \) to \( n \). It is used to count the number of ways to arrange \( n \) objects.
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are recursive, meaning that each factorial is built on the previous one:

• \( 1! = 1 \)
• \( 2! = 2 \times 1 \)
• \( 3! = 3 \times 2 \times 1 \)
• \( n! = n \times (n-1)! \)

Being comfortable with factorials helps in understanding more complex mathematical concepts like permutations and combinations, where they are frequently used in formulas.

Permutation formula

Permutations are essential in determining the arrangements of a set of items. The permutation formula \( P(n, r) \) calculates the number of ways to arrange \( r \) objects from a set of \( n \) objects where order matters.
The formula is expressed as:\[ P(n, r) = \frac{n!}{(n-r)!} \]Here, \( n! \) is the factorial of the total objects, while \((n-r)!\) accounts for the difference between the total objects and those being arranged.

• \( n \) is the total number of items.
• \( r \) is the number of items to arrange.
• Order of arrangement is crucial in permutations.

Using this formula, one can determine arrangements for different scenarios, such as word orderings, seating arrangements, and various sequential setups.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combinations of elements within a set. It involves various principles and formulas for systematically counting possible configurations, like permutations and combinations.
Some key aspects of combinatorics include:

• Arrangements vs. selections: Permutations handle arrangements where order matters, while combinations consider selections where order doesn't matter.
• Use of factorials: Factorials are fundamental in calculating permutations and combinations, as they provide a way to count possible arrangements efficiently.
• Practical applications: Combinatorics applies to fields like computer science, probability, and statistics, allowing verification of algorithms, prediction of outcomes, and more.

Understanding combinatorics can simplify tackling complex mathematical problems by breaking them into more manageable counts and arrangements.