/*! 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 14 Find the value of each permutati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 14

Find the value of each permutation. $$_7 P_{7}$$

Short Answer

Expert verified
The value of \(_7 P_{7}\) is 5040.

Step by step solution

01

Understand the Permutation Formula

Permutations are calculated using the formula }}
02

Apply the Permutation Formula

Substitute the given values into the formula: \(n = 7\) and \(r = 7\). Thus, the expression becomes $$\frac{7!}{(7-7)!}$$.
03

Simplify the Factorials

First, compute the factorials. $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040.$$ And \(0! = 1\) by definition. Plug these values back into the formula: $$\frac{7!}{0!} = 5040.$$.
04

Compute the Final Value

Perform the division: $$\frac{5040}{1} = 5040.$$.

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.

Factorial
In mathematics, understanding factorials is key to grasping more complex topics like permutations. A factorial, denoted as \( n! \), is the product of all positive integers up to the number \( n \).
For example:
  • \( 3! = 3 \times 2 \times 1 = 6 \)
  • \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
  • \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

The special case \( 0! \) is defined to be 1 by convention. This is important in many mathematical formulas and ensures they work correctly for all values of \( n \). When calculating permutations, you will frequently encounter factorials, so it's essential to practice recognizing and computing them efficiently.
Permutation Formula
Permutations deal with the arrangement of objects in a specific order. The permutation formula is useful for problems where the order in which objects are arranged is important.
The general formula for the number of permutations of \( n \) objects taken \( r \) at a time is given by:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
This formula calculates how many different ways you can arrange \( r \) objects out of a total of \( n \) objects.
To see this in action, let's use the example from the exercise with \( n = 7 \) and \( r = 7 \):
  • Substitute \( n \) and \( r \) into the formula: \[ P(7, 7) = \frac{7!}{(7-7)!} = \frac{7!}{0!} \]
  • We already know that \( 7! = 5040 \) and \( 0! = 1 \) by definition. So, \[ \frac{5040}{1} = 5040 \]

This demonstrates that there are 5040 different ways to arrange 7 objects taken 7 at a time.
Combinatorics
Combinatorics is a branch of mathematics focusing on counting, arranging, and combination of objects. Understanding combinatorics is essential for solving complex problems in probability, statistics, and computer science.
Key concepts include:
  • Permutations: The number of ways to arrange a set of objects where order matters.
  • Combinations: The number of ways to choose objects from a set where order does not matter. The formula for combinations of \( n \) objects taken \( r \) at a time is \ \frac{n!}{r!(n-r)!} \

When working with permutations and combinations, practice visualizing different ways items can be arranged or selected. This helps deepen your understanding and makes solving these problems much easier.
By mastering these fundamental concepts, you'll be well-equipped to tackle more advanced topics in mathematics and beyond.

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

Determine whether the following probabilities are computed using classical methods, empirical methods, or subjective methods. (a) The probability of having eight girls in an eight-child family is \(0.390625 \%\) (b) On the basis of a study of families with eight children, the probability of a family having eight girls is \(0.54 \%\) (c) According to a sports analyst, the probability that the Chicago Bears will win their next game is about \(30 \%\). (d) On the basis of clinical trials, the probability of efficacy of a new drug is \(75 \%\)

Bob is asked to construct a probability model for rolling a pair of fair dice. He lists the outcomes as \(2,3,4,5,6,7,8,9,10,11,12 .\) Because there are 11 outcomes, he reasoned, the probability of rolling a two must be \(\frac{1}{11} .\) What is wrong with Bob's reasoning?

Suppose you are dealt 5 cards from a standard 52 -card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52 card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4 th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

True or False: In a probability model, the sum of the probabilities of all outcomes must equal 1.

Find the probability of the indicated event if \(P(E)=0.25\) and \(P(F)=0.45\) Find \(P(E \text { or } F)\) if \(E\) and \(F\) are mutually exclusive.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete 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.