/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Find the value of each of the fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 40

Find the value of each of the following using the appropriate formula. \(\begin{array}{cccccccc}6 ! & 11 ! & (7-2) ! & (15-5) ! & { }_{8} C_{2} & { }_{5} C_{0} & { }_{5} C_{5} & { }_{6} C_{4} & { }_{11} C_{7} & { }_{9} P_{6} & { }_{12} P_{8}\end{array}\)

Short Answer

Expert verified
The results are: 1. For factorials: \(6! = 720, 11! = 39916800, (7-2)! = 120, (15-5)! = 3628800\). 2. For combinations: \(_{8}C_{2} = 28, _{5}C_{0} = 1, _{5}C_{5} = 1, _{6}C_{4} = 15, _{11}C_{7} = 330\). 3. For permutations: \(_{9}P_{6} = 60480, _{12}P_{8} = 19958400\).

Step by step solution

01

Factorial Calculations

The factorial of a non-negative integer number \(n\), denoted \(n!\), is the product of all positive integers less than or equal to \(n\). 鈥� For \(6!\), calculate \(6 * 5 * 4 * 3 * 2 * 1 = 720\).鈥� For \(11!\), calculate \(11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 39916800\).鈥� For \((7-2)!\), calculate the factorial of the result of \(7 - 2 = 5\) yielding \(5 * 4 * 3 * 2 * 1 = 120\).鈥� For \((15-5)!\), calculate the factorial of the result of \(15 - 5 = 10\) yielding \(10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3628800\).
02

Combination Calculations

Combination \(_{n}C_{r}\) of a set is given by the formula \(_{n}C_{r} = \frac{n!}{r!*(n-r)!}\).鈥� For \(_{8}C_{2}\), calculate \(\frac{8!}{2!(8-2)!} = 28\).鈥� For \(_{5}C_{0}\), calculate \(\frac{5!}{0!(5-0)!} = 1\).鈥� For \(_{5}C_{5}\), calculate \(\frac{5!}{5!(5-5)!} = 1\).鈥� For \(_{6}C_{4}\), calculate \(\frac{6!}{4!(6-4)!} = 15\).鈥� For \(_{11}C_{7}\), calculate \(\frac{11!}{7!(11-7)!} = 330\).
03

Permutation Calculations

Permutation \(_{n}P_{r}\) of a set is given by the formula \(_{n}P_{r} = \frac{n!}{(n-r)!}\).鈥� For \(_{9}P_{6}\), calculate \(\frac{9!}{(9-6)!} = 60480\).鈥� For \(_{12}P_{8}\), calculate \(\frac{12!}{(12-8)!} = 19958400\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Combinations
Combinations are a way to choose items from a larger set when the order of choices does not matter. This concept is widely used in math, especially in statistics and probability. Let鈥檚 say you have a deck of cards and you want to find how many ways you can select a group of 3 cards. That is a classic combination problem.

Mathematically, combinations are calculated using the formula:
  • \( _{n}C_{r} = \frac{n!}{r!(n-r)!} \)
where \(n\) is the total number of items, and \(r\) is the number of items to choose. In simpler terms, it is the number of ways to choose \(r\) items from \(n\) items.

Some key points about combinations:
  • The order of selection does not matter.
  • If \(r = 0\) or \(r = n\), there is only one way to choose items, either none (an empty set) or all of them.
  • The combination formula can be reduced if \(r > n-r\) by simplifying calculations using smaller numbers.
Permutations
Permutations are related to combinations but with a significant twist: the order of items matters in permutations. Think of permutations as arranging books on a bookshelf where the sequencing of books affects the outcome.

The formula to determine permutations is:
  • \( _{n}P_{r} = \frac{n!}{(n-r)!} \)
where \(n\) is the total number of items to pick from, and \(r\) is the number of items to arrange.

Important features of permutations:
  • The sequence of arrangements is important. \(ABC\) and \(CAB\) are considered different in permutations.
  • If all items are to be arranged, i.e., \(r = n\), then the permutation is simply \(n!\).
  • The number of permutations is generally greater than the number of combinations due to the importance of order.
Mathematical Formulas
Mathematical formulas simplify the process of arranging and selecting items. These are expressions that encapsulate a set of instructions, translating them into calculations that can be performed with ease.

For example, factorial notation \(n!\) is used in both permutations and combinations to compute the total number of possibilities. It is defined as the product of all integers from 1 to \(n\).

The combination formula illustrates choosing without concern for order, while the permutation formula accounts for different orders.
In summary:
  • Factorials help in reducing large multiplication operations.
  • Combinations focus on selection, disregarding arrangement.
  • Permutations emphasize both selection and arrangement.
Integer Calculations
Integer calculations form the backbone of evaluating mathematical formulas involving factorial, combinations, and permutations. When dealing with such calculations, precision is key as even a small error can lead to incorrect results.

Here鈥檚 what you need to know about integer calculations:
  • Factorials: These involve multiplying a sequence of increasing integers, useful in permutations and combinations.
  • Addition and subtraction are often required to determine parts of calculations such as \(n-r\) in formulas.
  • Division: Used to render formulas like \( _{n}C_{r} \) and \( _{n}P_{r} \) into manageable calculations by canceling common terms.
Accurate and methodical integer calculations ensure exact results, which is fundamental when learning or applying advanced mathematical techniques.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Briefly explain the two characteristics (conditions) of the probability distribution of a discrete random variable.

The following table gives the probability distribution of a discrete random variable \(x\) $$ \begin{array}{l|lllllll} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & .11 & .19 & .28 & .15 & .12 & .09 & .06 \\ \hline \end{array} $$ Find the following probabilities. a. \(P(x=3)\) b. \(P(x \leq 2)\) c. \(P(x \geq 4)\) d. \(P(1 \leq x \leq 4)\) e. Probability that \(x\) assumes a value less than 4 f. Probability that \(x\) assumes a value greater than 2 g. Probability that \(x\) assumes a value in the interval 2 to 5

The following table, reproduced from Exercise \(5.12\), lists the probability distribution of the number of patients entering the emergency room during a 1-hour period at Millard Fellmore Memorial Hospital. $$ \begin{array}{l|ccccccc} \hline \text { Patients per hour } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Probability } & .2725 & .3543 & .2303 & .0998 & .0324 & .0084 & .0023 \\ \hline \end{array} $$ Calculate the mean and standard deviation for this probability distribution.

According to a March 25,2007 Pittsburgh Post-Gazette article, \(30 \%\) to \(40 \%\) of U.S. taxpayers cheat on their returns. Suppose that \(30 \%\) of all current U.S. taxpayers cheat on their returns. Use the binomial probabilities table (Table I of Appendix C) or technology to find the probability that the number of U.S. taxpayers in a random sample of 14 who cheat on their taxes is a. at least 8 \(\mathrm{b}\), at most 3 c. 3 to 7

A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are $$\$ 10$$ million from the office building, $$\$ 5$$ million from the theater, and $$\$ 2$$ million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are \(.15, .30, .45\), and 10, respectively. Let \(x\) be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of \(x\). Find the mean and standard deviation of \(x\). Give a brief interpretation of the values of the mean and standard deviation.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.