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Chapter 7: Problem 21

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

Short Answer

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

Step by step solution

01

Write down the given expression

We are given the expression: \(P(n, n-2)\)
02

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: \(P(n, n-2) = \frac{n!}{(n-(n-2))!}\)
03

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: \(P(n, n-2) = \frac{n!}{2!}\)
04

Compute the factorials

To finish the evaluation, we need to compute the factorials of n and 2: \(n! = n \cdot (n-1) \cdot (n-2)! \) \(2! = 2 \cdot 1 = 2\)
05

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: \( P(n, n-2) = \frac{n \cdot (n-1) \cdot (n-2)!}{2} \)

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Key Concepts

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

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.

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Most popular questions from this chapter

Fifty raffle tickets are numbered 1 through 50 , and one of them is drawn at random. What is the probability that the number is a multiple of 5 or 7 ? Consider the following "solution": Since 10 tickets bear numbers that are multiples of 5 and since 7 tickets bear numbers that are multiples of 7 , we conclude that the required probability is $$ \frac{10}{50}+\frac{7}{50}=\frac{17}{50} $$ What is wrong with this argument? What is the correct answer?

Suppose the probability that Bill can solve a problem is \(p_{1}\) and the probability that Mike can solve it is \(p_{2}\). Show that the probability that Bill and Mike working independently can solve the problem is \(p_{1}+p_{2}-p_{1} p_{2}\).

A leading manufacturer of kitchen appliances advertised its products in two magazines: Good Housekeeping and the Ladies Home Journal. A survey of 500 customers revealed that 140 learned of its products from Good Housekeeping, 130 learned of its products from the Ladies Home Journal, and 80 learned of its products from both magazines. What is the probability that a person selected at random from this group saw the manufacturer's advertisement in a. Both magazines? b. At least one of the two magazines? c. Exactly one magazine?

Let \(S\) be a sample space for an experiment. Show that if \(E\) is any event of an experiment, then \(E\) and \(E^{c}\) are mutually exclusive.

In a television game show, the winner is asked to select three prizes from five different prizes, \(A, B\), \(\mathrm{C}, \mathrm{D}\), and \(\mathrm{E} .\) a. Describe a sample space of possible outcomes (order is not important). b. How many points are there in the sample space corresponding to a selection that includes A? c. How many points are there in the sample space corresponding to a selection that includes \(\mathrm{A}\) and \(\mathrm{B}\) ? d. How many points are there in the sample space corresponding to a selection that includes either \(\mathrm{A}\) or \(\mathrm{B}\) ?
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