/*! 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 18 Evaluate the given expression. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 18

Evaluate the given expression. $$ C(10,3) $$

Short Answer

Expert verified
The given expression is \(C(10,3)\), which represents the number of possible combinations when selecting 3 items from a set of 10. Using the combination formula, we have \(C(10, 3) = \frac{10!}{3!(10-3)!}\), which simplifies to \(C(10, 3) = \frac{3,628,800}{30,240}\). The final result is 120 possible combinations.

Step by step solution

01

Identify n and r

In this case, the given expression is C(10,3), so we have n = 10 and r = 3.
02

Factorial calculations

Calculate the factorials of n, r, and (n-r): - n! = 10! = 3,628,800 - r! = 3! = 6 - (n-r)! = (10-3)! = 7! = 5,040
03

Apply the combination formula

Insert the values of n, r, and their respective factorials into the combination formula: \[C(10, 3) = \frac{10!}{3!(10-3)!}\]
04

Evaluate the expression

Substitute the factorial values found in step 2 and evaluate the expression: \[C(10, 3) = \frac{3,628,800}{6 \times 5,040} = \frac{3,628,800}{30,240}\]
05

Simplify the result

Divide the numbers to obtain the final result: \[C(10, 3) = 120\] The result is 120 possible combinations when selecting 3 items from a set of 10.

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.

Combination Formula
In combinatorics, combinations are an essential concept used to find the number of possible ways to choose a subset of items from a larger set without regards to the order of selection. The formula for combinations is denoted by \( C(n, r) \), which represents choosing \( r \) items from \( n \) total items. The mathematical formula is written as:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n! \) denotes the factorial of \( n \), \( r! \) the factorial of \( r \), and \( (n-r)! \) the factorial of the difference between \( n \) and \( r \).
  • This formula is pivotal when the order does not matter, distinguishing it from permutations where the order is important.
  • Understanding how to apply the combination formula provides valuable tools for calculating probabilities and analyzing statistical data.
Factorials
Factorials are fundamental in combinatorics and mathematical calculations. When you see the symbol \( n! \), it means the product of all positive integers from 1 to \( n \). For example:
  • \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
  • \( 3! = 3 \times 2 \times 1 = 6 \)
Factorials grow quickly with increasing \( n \). They help in calculating combinations and permutations by accounting for all possible arrangements of a set.
Keep in mind:
  • \( 0! \) is defined as 1, which is a special case for factorials.
  • Factorials are used in various fields, including probability, algebra, and calculus.
Permutations
Unlike combinations, permutations consider the order of items. In permutations, each different arrangement counts as a distinct possibility. For permutations of \( n \) items taken \( r \) at a time, the formula is:\[ P(n, r) = \frac{n!}{(n-r)!} \]highlighting how it differs from the combination formula. For example:
  • Arranging 3 out of 5 books on a shelf is a permutation problem.
  • If the order were not important, it would become a combination problem.
Therefore, permutations are ideal when the sequence is crucial, such as in arrangements, rankings, or sequences.

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

According to a study conducted in 2003 concerning the participation, by age, of \(401(\mathrm{k})\) investors, the following data were obtained: $$ \begin{array}{lccccc} \hline \text { Age } & 20 \mathrm{~s} & 30 \mathrm{~s} & 40 \mathrm{~s} & 50 \mathrm{~s} & 60 \mathrm{~s} \\ \hline \text { Percent } & 11 & 28 & 32 & 22 & 7 \\ \hline \end{array} $$ a. What is the probability that a \(401(\mathrm{k})\) investor selected at random is in his or her 20 s or 60 s? b. What is the probability that a \(401(\mathrm{k})\) investor selected at random is under the age of 50 ?

List the simple events associated with each experiment. In a survey conducted to determine whether movie attendance is increasing \((i)\), decreasing \((d)\), or holding steady \((s)\) among various sectors of the population, participants are classified as follows: Group 1: Those aged 10-19 Group 2: Those aged 20-29 Group 3: Those aged 30-39 Group 4: Those aged 40-49 Group 5: Those aged 50 and older The response and age group of each participant are recorded.

A certain airport hotel operates a shuttle bus service between the hotel and the airport. The maximum capacity of a bus is 20 passengers. On alternate trips of the shuttle bus over a period of \(1 \mathrm{wk}\), the hotel manager kept a record of the number of passengers arriving at the hotel in each bus. a. What is an appropriate sample space for this experiment? b. Describe the event \(E\) that a shuttle bus carried fewer than ten passengers. c. Describe the event \(F\) that a shuttle bus arrived with a full load.

In a survey of 106 senior information technology and data security professionals at major U.S. companies regarding their confidence that they had detected all significant security breaches in the past year, the following responses were obtained. $$ \begin{array}{lcccc} \hline & \begin{array}{c} \text { Very } \\ \text { Answer } \end{array} & \begin{array}{c} \text { Moderately } \\ \text { confident } \end{array} & \begin{array}{c} \text { Not very } \\ \text { confident } \end{array} & \begin{array}{c} \text { Not at all } \\ \text { confident } \end{array} & \text { confident } \\ \hline \text { Respondents } & 21 & 56 & 22 & 7 \\ \hline \end{array} $$ What is the probability that a respondent in the survey selected at random a. Had little or no confidence that he or she had detected all significant security breaches in the past year? b. Was very confident that he or she had detected all significant security breaches in the past year?

In a survey of 200 employees of a company regarding their \(401(\mathrm{k})\) investments, the following data were obtained: 141 had investments in stock funds. 91 had investments in bond funds. 60 had investments in money market funds. 47 had investments in stock funds and bond funds. 36 had investments in stock funds and money market funds. 36 had investments in bond funds and money market funds. 22 had investments in stock funds, bond funds, and money market funds. What is the probability that an employee of the company chosen at random a. Had investments in exactly two kinds of investment funds? b. Had investments in exactly one kind of investment fund? c. Had no investment in any of the three types of funds?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.