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Chapter 4: Problem 14

Compute \(P_{8,3}\).

Short Answer

Expert verified
\(P_{8,3} = 336\).

Step by step solution

01

Understand the Formula for Permutations

The notation \(P_{n,r}\) refers to the number of permutations of \(n\) items taken \(r\) at a time. The formula is \(P_{n,r} = \frac{n!}{(n-r)!}\). This formula helps find how many different ways you can arrange \(r\) items out of \(n\) items.
02

Identify Values of n and r

For \(P_{8,3}\), we have \(n = 8\) and \(r = 3\). This means we want to find out how many ways we can arrange 3 items from a total of 8.
03

Calculate the Factorial of n

Calculate \(8!\) which is the factorial of 8. \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\).
04

Calculate the Factorial of (n-r)

Calculate \((8-3)!\) which is \(5!\). \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
05

Apply the Permutation Formula

Substitute the factorials into the permutation formula: \(P_{8,3} = \frac{8!}{(8-3)!} = \frac{40320}{120}\).
06

Compute the Division

Divide \(40320\) by \(120\) to get \(336\). So, \(P_{8,3} = 336\).

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Key Concepts

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

Factorial
In mathematics, the factorial of a non-negative integer is the product of all positive integers less than or equal to that integer. It's a key operation used in permutations and combinations. The symbol for factorial is an exclamation mark (!). For instance, when you see \( n! \), it represents the factorial of \( n \). Let鈥檚 break down this concept further:
  • Basics: By definition, \( n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 \). It's important to remember that \( 0! = 1 \) as a special case by definition.
  • Examples: If \( n = 5 \), then \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This calculation becomes crucial when dealing with permutations.
Factorials grow very quickly with larger numbers, which is why they're a perfect tool to calculate permutations and combinations, helping us determine all possible arrangements or selections.
Permutation Formula
The permutation formula is a foundational concept in combinatorics used to calculate the number of possible arrangements of a subset of items from a set. Given a set of \( n \) items, a permutation is an ordered arrangement of \( r \) items taken from that set. The formula is expressed as:\[P_{n,r} = \frac{n!}{(n-r)!}\]Here鈥檚 what this formula tells us:
  • Understanding the Terms: \( n \) is the total number of items, and \( r \) is the number of items you want to arrange.
  • Application: We apply \( P_{n,r} \) to determine how many different ways we can organize those \( r \) items among all \( n \) items.
  • Example: If you have 8 books and want to know in how many ways you can arrange 3 of them, you would use the permutation formula \( P_{8,3} \).
Using the permutation formula, we help break down complex organizational problems into manageable calculations.
Combinatorics
Combinatorics is the area of mathematics that studies how to count, arrange, and combine objects according to defined rules. It's vast and includes permutations, combinations, and more. The motivation behind combinatorics is understanding all possible arrangements or selections of items.
  • Permutations vs. Combinations: While permutations are about arranging items in order, combinations focus on selecting items without regard to order.
  • Practical Uses: In everyday problem-solving, combinatorics can be applied to logistical management, computer algorithms, and statistical analysis.
By mastering combinatorics, you improve your problem-solving skills and mathematical thinking, allowing for effective strategies in both academic and real-world scenarios.

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Most popular questions from this chapter

You roll two fair dice, a green one and a red one. (a) What is the probability of getting a sum of 6 ? (b) What is the probability of getting a sum of 4 ? (c) What is the probability of getting a sum of 6 or 4? Are these outcomes mutually exclusive?

Diagnostic tests of medical conditions can have several types of results. The test result can be positive or negative, whether or not a patient has the condition. A positive test \((+)\) indicates that the patient has the condition. A negative test \((-)\) indicates that the patient does not have the condition. Remember, a positive test does not prove that the patient has the condition. Additional medical work may be required. Consider a random sample of 200 patients, some of whom have a medical condition and some of whom do not. Results of a new diagnostic test for the condition are shown. $$\begin{aligned} &\text { Results of a new diagnostic test for the condition are shown. }\\\ &\begin{array}{lccc} \hline & \begin{array}{c} \text { Condition } \\ \text { Present } \end{array} & \begin{array}{c} \text { Condition } \\ \text { Absent } \end{array} & \text { Row Total } \\ \hline \text { Test Result }+ & 110 & 20 & 130 \\ \text { Test Result }- & 20 & 50 & 70 \\ \text { Column Total } & 130 & 70 & 200 \\ \hline \end{array} \end{aligned}$$ Assume the sample is representative of the entire population. For a person selected at random, compute the following probabilities: (a) \(P(+\mid\) condition present \()\); this is known as the sensitivity of a test. (b) \(P(-\mid\) condition present \() ;\) this is known as the false-negative rate. (c) \(P(-\mid\) condition absent \() ;\) this is known as the specificity of a test. (d) \(P(+\mid\) condition absent \()\); this is known as the false-positive rate. (e) \(P\) (condition present and \(+\) ); this is the predictive value of the test. (f) \(P\) (condition present and -).

If two events are mutually exclusive, can they occur concurrently? Explain.

Based on data from the Statistical Abstract of the United States, 112 th Edition, only about \(14 \%\) of senior citizens \((65\) years old or older) get the flu each year. However, about \(24 \%\) of the people under 65 years old get the flu each year. In the general population, there are \(12.5 \%\) senior citizens \((65\) years old or older). (a) What is the probability that a person selected at random from the general population is a senior citizen who will get the flu this year? (b) What is the probability that a person selected at random from the general population is a person under age 65 who will get the flu this year? (c) Answer parts (a) and (b) for a community that has \(95 \%\) senior citizens. (d) Answer parts (a) and (b) for a community that has \(50 \%\) senior citizens.

Consider a series of events. How does a tree diagram help you list all the possible outcomes of a series of events? How can you use a tree diagram to determine the total number of outcomes of a series of events?
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