/*! 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 67 A shipment of 25 television sets... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 67

A shipment of 25 television sets contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at least two good units?

Short Answer

Expert verified
Therefore, the number of ways for each scenario are: (a) for all good units, calculate \( \binom{22}{4} \), (b) for two good units, calculate \( \binom{22}{2} \times \binom{3}{2} \), (c) for at least two good units, calculate the sum of \( \binom{22}{2} \times \binom{3}{2} \), \( \binom{22}{3} \times \binom{3}{1} \), and \( \binom{22}{4} \)

Step by step solution

01

Find Total Number of Ways to Select Four Units

There are 25 units in total and the company wants to purchase four units. The total number of ways to select four units out of 25, not considering whether the units are defective or not, can be found using the combination formula: \( \binom{25}{4} = \frac{25!}{4!(25-4)!} \)
02

Find Number of Ways to Select All Good Units

There are 22 good units (25 - 3 defective units). The number of ways to select all good units can be found by substituting 'n' with 22 (total number of good units) and 'r' with 4 (units to purchase) in the combination formula: \( \binom{22}{4} = \frac{22!}{4!(22-4)!} \)
03

Find Number of Ways to Select Two Good Units

Here the company wants to select two good units and two defective units. The number of ways to select two good units out of 22 can be found using the combination formula: \( \binom{22}{2} = \frac{22!}{2!(22-2)!} \). Similarly, the number of ways to select two defective units out of 3 can be found using the same formula: \( \binom{3}{2} = \frac{3!}{2!(3-2)!} \). As we want two good and two defective units, multiply these two results to get the total.
04

Find Number of Ways to Select At Least Two Good Units

In this scenario, there can be 2, 3, or 4 good units. To compute this, find the number of ways for each of these cases and then sum the results. The number of ways to select 2 good and 2 defective units has been calculated in the previous step. For 3 good and 1 defective, use the similar method: \( \binom{22}{3} \times \binom{3}{1} \), and for 4 good units it has been calculated in the step related to 'all good units'. Sum these three sets of results to get the total.

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!

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

In Exercises 25 - 30, find the probability for the experiment of tossing a six-sided die twice. The sum is odd and no more than \( 7 \).

In Exercises 25 - 30, find the probability for the experiment of tossing a six-sided die twice. The sum is odd or prime

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = n (n - 1)(n - 2)\)

In Exercises 131-134, use the following definition of the arithmetic mean \( \bar{x} \) of a set of \( n \) measurements \( x_1, x_2, x_3, \dots , x_n \). \( \displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \) Find the arithmetic mean of the six checking account balances \( \$327.15, \$785.69, \$433.04, \$265.38, \$604.12, \) and \( \$590.30 \). Use the statistical capabilities of a graphing utility to verify your result.

In Exercises 103-112, use sigma notation to write the sum. \( 3 - 9 + 27 - 81 + 243 - 729 \)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.