/*! 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 34 A bag contains three red marbles... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 34

A bag contains three red marbles, two green ones, one lavender one, two yellows, and two orange marbles. How many sets of three marbles include none of the yellow ones?

Short Answer

Expert verified
There are 56 sets of three marbles that do not include any yellow marbles.

Step by step solution

01

Determine the total number of marbles

The given bag contains: - 3 red marbles - 2 green marbles - 1 lavender marble - 2 yellow marbles - 2 orange marbles This sums up to \(10\) marbles in total.
02

Calculate the total number of sets of 3 marbles

Use combinations (denoted as C(n, k)) to find the total number of ways to choose 3 marbles out of 10: \(C(10, 3) = \frac{10!}{3! (10-3)!} = \frac{10!}{3! 7!} = 120\) There are 120 possible sets of three marbles.
03

Calculate the number of sets including yellow marbles

There are two yellow marbles in the bag. We want to find the number of sets of three marbles that include at least one yellow marble. There are two possibilities: Case 1: Exactly one yellow marble in the 3-marble set: \(C(2, 1)\ 脳 C(8, 2) = 2 脳 28 = 56\) Case 2: Exactly two yellow marbles in the 3-marble set: \(C(2, 2)\ 脳 C(8, 1) = 1 脳 8 = 8\) Now, add the possibilities together to find the total number of sets including yellow marbles: \(56 + 8 = 64\) There are 64 sets of three marbles which include at least one yellow marble.
04

Calculate the number of sets without yellow marbles

To find the number of sets of three marbles without yellow marbles, subtract the number of sets including yellow marbles (found in Step 3) from the total number of sets of three marbles (found in Step 2): \(120 - 64 = 56\) There are 56 sets of three marbles that do not include any yellow marbles.

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

Evaluate each number. $$ C(100,97) $$

How many three-letter (unordered) sets are possible that use the letters \(\mathrm{q}, \mathrm{u}, \mathrm{a}, \mathrm{k}, \mathrm{e}, \mathrm{s}\) at most once each?

Use Venn diagrams to illustrate the following identities for subsets \(A, B\), and \(\operatorname{Cof} S .\) $$ A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \quad \text { Distributive Law } $$

Ice Cream At the beginning of 2002, Baskin-Robbins claimed to have "nearly 1,000 different ice cream flavors." \(^{\prime \prime}\) Assuming that you could choose from 1,000 different flavors, that you could have your ice cream in a cone, a cup, or a sundae, and that you could choose from a dozen different toppings, how many different desserts could you have?

Use Venn diagrams to illustrate the following identities for subsets \(A, B\), and \(\operatorname{Cof} S .\) $$ \begin{aligned} &(A \cap B) \cap C=A \cap(B \cap C)\\\ &\text { Associative Law } \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.