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Chapter 14: Problem 16

If the subject guesses two shapes correctly and two incorrectly, in how many ways can you arrange the sequence of correct and incorrect guesses? a. \(\left(\begin{array}{l}3 \\ 2\end{array}\right)=3\) b. \(\left(\begin{array}{l}4 \\ 2\end{array}\right)=12\) c. \(\left(\begin{array}{l}4 \\ 2\end{array}\right)=6\)

Short Answer

Expert verified
There are 6 ways, option (c) is correct.

Step by step solution

01

Understand the Problem

We need to find the number of ways to arrange two correct guesses and two incorrect guesses. This is essentially a permutation problem where we have 4 positions to fill with 2 correct guesses and 2 incorrect guesses.
02

Identify Total Positions and Choices

There are 4 total positions for the guesses. We must choose 2 of these positions to be correct guesses, which means the remaining 2 will automatically be incorrect guesses.
03

Apply Combination Formula

Use the combination formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \),where \(n\) is the total number of items to choose from (in this case, 4) and \(k\) is the number of items to choose (2 correct guesses).
04

Calculate Specific Value

Substitute 4 for \(n\) and 2 for \(k\): \[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]
05

Evaluate Given Options

Compare the calculated value with the given multiple-choice answers. The correct answer among options is (c) \( \binom{4}{2} = 6 \).

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Key Concepts

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

Binomial Coefficient
In combinatorics, the binomial coefficient is a fundamental concept used to determine the number of ways to choose a subset of items from a larger set. It can be represented as \( \binom{n}{k} \), pronounced "n choose k." Here, \( n \) is the total number of items, and \( k \) is the number of items to choose.

Key aspects of binomial coefficients include:
  • **Symmetry:** The binomial coefficient is symmetric, meaning \( \binom{n}{k} = \binom{n}{n-k} \). This reflects the fact that choosing \( k \) items from \( n \) is the same as leaving out \( n-k \) items.
  • **Pascal's Triangle:** Each number in Pascal's Triangle represents a binomial coefficient, where each number is the sum of the two numbers directly above.This diagram helps visualize combinatorial relationships.
  • **Applications:** Besides arranging guesses, binomial coefficients are used in probability, statistics, and algebra, such as in the Binomial Theorem for expanding polynomial expressions.
Calculating binomial coefficients involves using factorials, making them a valuable tool for solving combinatorics problems effectively.
Permutation
Permutations involve arranging a set of items in a distinct order. Unlike combinations, permutations consider the sequence in which the items are arranged, making order very important.

Some important features of permutations are:
  • **Formula:** The number of permutations of \( n \) distinct items is \( n! \) (n factorial). This means multiplying all whole numbers from \( n \) down to 1.
  • **Different Types:** There are permutations with and without repetition. In exercises where items must not repeat, you use \( n! \). Allowing repetition changes the calculation, often using \( n^r \), where \( r \) is the number of positions to fill.
  • **Importance in Probability and Puzzles:** Permutations are important in calculating probabilities and solving problems like anagrams and puzzle arrangements, where order is crucial.
Understanding permutations helps decide when order matters in a problem, distinguishing them from combinations, where order doesn鈥檛 matter.
Combination Formula
The combination formula is used when the order of selection does not matter. In scenarios where you just need to choose items, rather than arrange them in a sequence, combinations are used. This is particularly true in lottery selections or course combinations.

To calculate combinations, the formula is:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here,
  • **\( n \)** refers to the total number of items available.
  • **\( k \)** is the number of items to be chosen.
  • **Factorials:** A factorial, represented by \( n! \), is the product of all positive integers up to \( n \).
For example, in our exercise, we used \( \binom{4}{2} \) to determine how many ways we could choose 2 correct guesses out of 4 guesses. The result was 6.

Combinations are essential in fields such as statistics and genetics, where the specific order of items is unimportant, focusing only on the selected group.

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Most popular questions from this chapter

Response Rates for Random Digit Dialing. When an opinion poll uses random digit dialing to select respondents for polls, the response rate is approximately \(10 \%\) for people contacted by cell phone. You watch a pollster dial 20 cell numbers many times using random digit dialing. a. What is the mean number of calls that yield a response? b. What is the standard deviation \(\sigma\) of the count of calls that yield a response? c. Suppose that the probability of getting a response were \(p=0.05\). How does this new \(p\) affect the standard deviation? What would be the standard deviation if \(p=0.01\) ? What does your work show about the behavior of the standard deviation of a binomial distribution as the probability of success gets closer to zero?

Multiple-Choice Tests. Here is a simple probability model for multiple-choice tests. Suppose each student has probability \(p\) of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher \(p\) than a weak student.) Answers to different questions are independent. a. Stacey is a good student for whom \(p=0.75\). Use the Normal approximation to find the probability that Stacey scores between \(70 \%\) and \(80 \%\) on a 100 -question test. b. If the test contains 250 questions, what is the probability that Stacey will score between \(70 \%\) and \(80 \%\) ? You see that Stacey's score on the longer test is more likely to be close to her "true score."

Response Rates for Random Digit Dialing. When an opinion poll uses random digit dialing to select respondents for polls, the response rate is approximately \(10 \%\) for people contacted by cell phone. You watch a pollster dial cell numbers that have been selected in this manner. \(X\) is the number of calls dialed before the pollster obtains a second response to the poll.

Hyundai Sales in 2018. Hyundai Motor America sold 677,946 vehicles in the United States in 2018, with the U.S.-built Elantra leading sales, with 200,415 cars sold. The other topselling nameplates in 2018 were the Tucson, with 135,348 sold, the Santa Fe, with 123,989 sold, and the Sonata, with 105, 118. 10. The company wants to undertake a survey of 2018 Hyundai buyers to ask them about satisfaction with their purchase. a. What proportion of the Hyundais sold in 2018 were Elantras? b. If Hyundai plans to survey a simple random sample of 1000 Hyundai buyers, what are the expected number and standard deviation of the number of Elantra buyers in the sample? c. What is the probability Hyundai will get fewer than 300 Elantra buyers in the sample?

College A dmissions. A small liheral arts college in Ohio would like to have an entering class of 500 st udents next year. Past experience shows that about \(40 \%\) of the students admitted will decide to attend. The college is planning to admit 1250 students. Suppose that students make their decisions independently and that the probability is \(0.40\) that a randomly chosen student will accept the offer of admission. a. What are the mean and standard deviation of the number of students who accept the admissions offer from this college? b. Using the Normal approximation, what is the probability that the college gets more students than it wants? Check that you can safely use the approximation. c. Use software or an online binomial calculator to compute the exact probability that the college gets more students than it wants. How good is the approximation in part (b)? d. To decrease the probability of getting more students than are wanted, does the college need to increase or decrease the number of students it admits? Using software or an online binomial calculator, what is the largest number of students that the college can admit if administrators want the exact probability of getting more students than they want to be no larger than \(5 \%\) ?
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