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Chapter 2: Problem 44

Show that \(\left(\begin{array}{c}n \\\ k\end{array}\right)=\left(\begin{array}{c}n \\ n-k\end{array}\right)\). Give an interpretation involving subsets.

Short Answer

Expert verified
The binomial coefficients are equal because choosing \( k \) elements equals choosing which \( n-k \) elements to exclude.

Step by step solution

01

Understanding the Binomial Coefficient

The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose \( k \) elements from \( n \) elements. It can also be interpreted as the number of different subsets of \( k \) elements that can be chosen from a larger set of \( n \) elements.
02

Applying the Formula

The formula for the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). This formula calculates the number of ways to pick \( k \) elements out of \( n \), considering that order does not matter, and each choice is made without replacement.
03

Express the Complementary Coefficient

Now consider the binomial coefficient \( \binom{n}{n-k} \). We can express this using the formula as \( \frac{n!}{(n-k)!k!} \). Notice that the denominators in \( \binom{n}{k} \) and \( \binom{n}{n-k} \) are identical, since \( k!(n-k)! = (n-k)!k! \).
04

Simplifying and Comparing

Since \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) and \( \binom{n}{n-k} = \frac{n!}{(n-k)!k!} \) have the same formula, it follows that \( \binom{n}{k} = \binom{n}{n-k} \). This equality holds because the order of multiplication in the denominator does not matter, proving the original statement.
05

Subsets Interpretation

The coefficients \( \binom{n}{k} \) and \( \binom{n}{n-k} \) are equal because choosing \( k \) elements to form a subset is equivalent to choosing \( n-k \) elements to leave out. Thus, selecting \( k \) elements from \( n \) is the same as selecting the \( n-k \) elements that are not included, showing both choices yield identical counts of sets.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Coefficient
A key concept in combinatorics is the binomial coefficient, commonly denoted as \( \binom{n}{k} \). This represents the number of ways to choose \( k \) elements from a set of \( n \) elements. It assumes all selections are made without considering order and without replacement. The binomial coefficient is fundamental in problems involving counting and can also be connected to other mathematical concepts, such as Pascal's Triangle and binomial expansions.
  • Expression: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
  • Meaning: Quantifies the number of different ways to form subsets of \( k \) elements from an overall set of \( n \) elements.
Understanding the binomial coefficient's calculation and significance provides a foundation for deeper exploration into the relationships between different combinations of elements.
Subsets
The idea of subsets is central in combinatorics and involves selecting a smaller group from a larger set. If you have a set with \( n \) elements, a subset is any selection of zero or more elements from that set.
  • Total Subsets: \( 2^n \) - as each element can either be included or excluded independently.
  • Subset of size \( k \): \( \binom{n}{k} \) - represents choosing exactly \( k \) items from \( n \).
Imagine a basket of fruits containing apples, oranges, and bananas. A subset of this basket might only contain apples and oranges. Recognizing subsets helps in understanding how these groups form and interact, offering a framework to solve more complex combinatorial problems.
Permutations
In contrast to combinatorial selections where item order doesn't matter, permutations focus on arrangements where order does count. For a set of \( n \) elements, a permutation involves arranging some or all of these elements in a specific sequence.
  • Common example: Arranging three books on a shelf.
  • Formula: The number of permutations of \( n \) distinct objects is \( n! \).
  • Partial permutations: The number of ways to sequence \( k \) objects from \( n \) is \( \frac{n!}{(n-k)!} \).
Understanding permutations is crucial when the sequence in which items are arranged affects the outcome, enriching our comprehension of both structured sequences and random arrangements.
n Choose k
The expression \( n \) choose \( k \) is a concise way to refer to the binomial coefficient \( \binom{n}{k} \). It signifies selecting \( k \) objects from a total of \( n \) without regard to the arrangement order.
  • Interpreted as choosing a team of \( k \) players from \( n \) available players.
  • Illustrates the same combinatorial principle both mathematically and conceptually through set theory.
The symmetry in its counting, such as \( \binom{n}{k} = \binom{n}{n-k} \), is highlighted through subset relationships: picking \( k \) elements to include is analogous to picking \( n-k \) elements to exclude. This demonstrates the equivalence of what is being selected and what is being left out, yielding the same result from different perspectives.

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Most popular questions from this chapter

The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group-blood group combinations. $$ \begin{array}{lccccc} &&&{\text { Blood Group }} \\ & & \text { O } & \mathbf{A} & \mathbf{B} & \mathbf{A B} \\ \hline {\text { Ethnic Group }} & \mathbf{1} & .082 & .106 & .008 & .004 \\ & \mathbf{2} & .135 & .141 & .018 & .006 \\ & \mathbf{3} & .215 & .200 & .065 & .020 \\ \hline \end{array} $$ Suppose that an individual is randomly selected from the population, and define events by \(A=\\{\) type A selected \(\\}, B=\\{\) type B selected \(\\}\), and \(C=\\{\) ethnic group 3 selected \(\\}\). a. Calculate \(P(A), P(C)\), and \(P(A \cap C)\). b. Calculate both \(P(A \mid C)\) and \(P(C \mid A)\), and explain in context what each of these probabilities represents. c. If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1?

A computer consulting firm presently has bids out on three projects. Let \(A_{i}=\\{\) awarded project \(i\\}\), for \(i=1,2,3\), and suppose that \(P\left(A_{1}\right)=.22, P\left(A_{2}\right)=.25, P\left(A_{3}\right)=.28\), \(P\left(A_{1} \cap A_{2}\right)=.11, P\left(A_{1} \cap A_{3}\right)=.05, P\left(A_{2} \cap A_{3}\right)=.07\), \(P\left(A_{1} \cap A_{2} \cap A_{3}\right)=.01\). Express in words each of the following events, and compute the probability of each event: a. \(A_{1} \cup A_{2}\) b. \(A_{1}^{\prime} \cap A_{2}^{\prime}\) c. \(A_{1} \cup A_{2} \cup A_{3}\) d. \(A_{1}^{\prime} \cap A_{2}^{\prime} \cap A_{3}^{\prime}\) e. \(A_{1}^{\prime} \cap A_{2}^{\prime} \cap A_{3}\) f. \(\left(A_{1}^{\prime} \cap A_{2}^{\prime}\right) \cup A_{3}\)

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk. $$ \begin{array}{lccc} \hline & \text { Small } & \text { Medium } & \text { Large } \\ \hline \text { Regular } & 14 \% & 20 \% & 26 \% \\ \text { Decaf } & 20 \% & 10 \% & 10 \% \\ \hline \end{array} $$ Consider randomly selecting such a coffee purchaser. a. What is the probability that the individual purchased a small cup? A cup of decaf coffee? b. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability? c. If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short (1-2 pages), medium (3-4 pages), or long (5-6 pages). Data on recent reviews indicates that \(60 \%\) of them are short, \(30 \%\) are medium, and the other \(10 \%\) are long. Reviews are submitted in either Word or LaTeX. For short reviews, \(80 \%\) are in Word, whereas \(50 \%\) of medium reviews are in Word and \(30 \%\) of long reviews are in Word. Suppose a recent review is randomly selected. a. What is the probability that the selected review was submitted in Word format? b. If the selected review was submitted in Word format, what are the posterior probabilities of it being short, medium, or long?

According to a July 31,2013 , posting on cnn.com subsequent to the death of a child who bit into a peanut, a 2010 study in the journal Pediatrics found that \(8 \%\) of children younger than 18 in the United States have at least one food allergy. Among those with food allergies, about \(39 \%\) had a history of severe reaction. a. If a child younger than 18 is randomly selected, what is the probability that he or she has at least one food allergy and a history of severe reaction? b. It was also reported that \(30 \%\) of those with an allergy in fact are allergic to multiple foods. If a child younger than 18 is randomly selected, what is the probability that he or she is allergic to multiple foods?
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