/*! 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 21 Evaluate each expression. Do not... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 21

Evaluate each expression. Do not use a calculator. $$C(6,0)$$

Short Answer

Expert verified
\( C(6,0) = 1 \)

Step by step solution

01

Understanding the Combination Formula

The combination formula is used to find the number of ways to choose a subset of items from a larger set, in which the order of the items does not matter. It is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items, and \( r \) is the number of items to choose. In this problem, \( n = 6 \) and \( r = 0 \).
02

Applying the Zero Combination Rule

According to the rule for combinations, choosing all 0 items from any number of items (\( n \)) results in exactly 1 way. Mathematically, this is represented by: \[ C(n, 0) = 1 \] Therefore, for our expression, \( C(6, 0) = 1 \).

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.

Factorials
Factorials are an important concept in mathematics and are denoted by an exclamation mark (!). They represent the product of all positive integers up to a given number. For example, if you want to find the factorial of 5, you calculate it as:\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]This operation is crucial when dealing with permutations and combinations, as it helps in determining the total possible arrangements of a set.
  • The factorial of 0 is a special case. By definition, it is equal to 1, i.e., \(0! = 1\).
  • Factorials grow rapidly with larger numbers, making them a powerful tool for counting.
Understanding how factorials work gives you insight into calculating combinations efficiently. They play a key role in the formula for combinations and permutations, providing the count of possible options.
Subset Selection
Subset selection refers to the process of choosing a smaller set of elements from a larger collection, where the order of the selection does not matter. This concept is often encountered in statistics and probability when dealing with questions like, "How many ways can we choose a few items from a group?" The formula used for subset selection through combinations is:\[C(n, r) = \frac{n!}{r! (n-r)!}\]where
  • \( n \) is the total number of items, and
  • \( r \) is the number of items to choose.
In our exercise example, when applying the formula with \( n = 6 \) and \( r = 0 \), you essentially calculate how many ways you can pick zero items from six. This is primarily about recognizing that the order does not matter in such selections.
Zero Combination Rule
The zero combination rule is a special rule in combinations that simplifies calculations significantly. It states that choosing zero elements from any set always results in exactly one possible combination, irrespective of the number of items in the set. Mathematically, it is expressed as:\[C(n, 0) = 1\]This is because there's only one way to select nothing from something: do not select anything at all. Some key points to remember:
  • It is always true that \( C(n, 0) = 1 \), highlighting the consistent nature of this concept across different scenarios.
  • This rule is efficient because it negates the need for lengthy calculations when zero items are to be selected.
Recognizing and applying the zero combination rule allows for quick solutions, especially when dealing with larger sets, where manually calculating factorials is not practical.

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

Use a formula to find the sum of each arithmetic series. $$89+84+79+74+\cdots+9+4$$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth. $$a_{n}=\sqrt{8} n+\sqrt{3}$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{12}=-108, a_{12}=-19$$

Color-Blind Males The probability that a male will be color blind is \(0.042 .\) Approximate the probabilities that in a group of 53 men, the following are true. A. Exactly 5 are color blind. B. No more than 5 are color blind. C. At least 1 is color blind.

The table gives the results of a survey of \(14,000\) college students who were cigarette smokers in a recent year. $$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Number of Cigarettes } \\\ \text { per Day } \end{array} & \begin{array}{c} \text { Percent } \\\ \text { (as a decimal) } \end{array} \\ \hline \text { Less than } 1 & 0.45 \\\ 1 \text { to } 9 & 0.24 \\ 10 \text { to } 19 & 0.20 \\ \text { A pack of } 20 \text { or more } & 0.11 \end{array}$$ Using the percents as probabilinies, approximate the probability that, out of 10 of these shudent smokers selected at random, the following were true. No more than 3 smoked less than 1 cigarette per day.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.