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Chapter 4: Problem 12

Express all probabilities as fractions. A survey with 12 questions is designed so that 3 of the questions are identical and 4 other questions are identical (except for minor changes in wording). How many different ways can the 12 questions be arranged?

Short Answer

Expert verified
There are 3,326,400 different ways to arrange the 12 questions.

Step by step solution

01

Understand the Problem

We need to determine the number of different ways to arrange 12 questions, where 3 questions are identical, and another 4 questions are also identical.
02

Use the Permutation Formula with Repetitions

When calculating permutations where some items are repeated, use the formula equation: \[P = \frac{n!}{n_1! \times n_2! \times ... \times n_k!}\]where \(n\) is the total number of items, and \(n_1, n_2, ..., n_k\) are the frequencies of each set of identical items. In this problem, we have 12 questions total (n), 3 identical questions (n1), and 4 other identical questions (n2).
03

Plug in the Values

Using the permutation formula from Step 2 and substituting \(n = 12\), \(n_1 = 3\), and \(n_2 = 4\), we get:\[P = \frac{12!}{3! \times 4!}\]
04

Calculate the Factorials

Calculate the factorials for 12, 3, and 4. Recall that \[n! = n \times (n-1) \times (n-2) \times ... \times 1\].So,\[12! = 479,001,600,3! = 6, 4! = 24\].
05

Compute the Final Answer

Substitute the values back into the formula:\[P = \frac{479,001,600}{6 \times 24} = \frac{479,001,600}{144} = 3,326,400\]

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Key Concepts

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

Factorials
When we talk about permutations and combinations, understanding factorials is essential. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a specific number. For example, the factorial of 4 (written as 4!) is calculated as:
4! = 4 脳 3 脳 2 脳 1 = 24
Factorials grow very quickly with larger numbers. In our exercise, we had to calculate the factorial of 12 (12!). This is a much larger number:
12! = 12 脳 11 脳 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 479,001,600
Factorials are essential when dealing with permutations because they help in computing the total number of possible arrangements.
Permutation Formula
The permutation formula is particularly helpful when some items are identical. The general permutation formula with repetitions is:
\[P = \frac{n!}{n_1! \times n_2! \times ... \times n_k!}\]
Here, 'n' is the total number of items, and 'n1, n2, ..., nk' are the frequencies of each repeating item. This formula adjusts the total number of permutations by dividing out the redundancies caused by the identical items. In the given problem, we applied this formula as follows: we had 12 total questions, with 3 identical questions and another set of 4 identical questions. Plugging these values into the formula, we got:
\[P = \frac{12!}{3! \times 4!}\]
After calculating each factorial, we arrive at the number of unique arrangements.
Identical Items
In permutations, handling identical items is crucial because identical items reduce the number of unique arrangements. For example, if you have 3 identical questions in a group of 12 questions, switching those 3 identical questions does not create a new unique arrangement.
In our exercise, we had two sets of identical items: 3 identical questions and 4 identical questions. If we ignored the identical nature of these items and simply used:\[12!\]
it would not give us the correct number of unique permutations because it would overcount the arrangements. Instead, by dividing by the factorials of the identical items (3! and 4!), we accurately account for the fact that swapping identical questions doesn't create a new permutation, achieving the precise number of unique arrangements:
\[P = \frac{12!}{3! \times 4!}\].
This way, we reduce the overcount and achieve the correct result which was 3,326,400 different ways to arrange the questions.

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Most popular questions from this chapter

Find the probability and answer the questions.Eye Color Each of two parents has the genotype brown/blue, which consists of the pair of alleles that determine eye color, and each parent contributes one of those alleles to a child. Assume that if the child has at least one brown allele, that color will dominate and the eyes will be brown. (The actual determination of eye color is more complicated than that.) a. List the different possible outcomes. Assume that these outcomes are equally likely. b. What is the probability that a child of these parents will have the blue/blue genotype? c. What is the probability that the child will have brown eyes?

Express all probabilities as fractions. Your professor has just collected eight different statistics exams. If these exams are graded in random order, what is the probability that they are graded in alphabetical order of the students who took the exam?

Express all probabilities as fractions. As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 75 and, in a separate drawing, you must also select the correct single number between 1 and \(15 .\) Find the probability of winning the jackpot. How does the result compare to the probability of being struck by lightning in a year, which the National Weather Service estimates to be \(1 / 960,000 ?\)

Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Brand Recognition The probability of randomly selecting an adult who recognizes the brand name of McDonald's is 0.95 (based on data from Franchise Advantage). Describe a procedure for using software or a TI- \(83 / 84\) Plus calculator to simulate the random selection of 50 adult consumers. Each individual outcome should be an indication of one of two results: (1) The consumer recognizes the brand name of McDonald's; (2) the consumer does not recognize the brand name of McDonald's.

$$\text { In Exercises } 5-8, \text { find the indicated complements.}$$ Flying In a Harris survey, adults were asked how often they typically travel on commercial flights, and it was found that \(P(N)=0.330,\) where \(N\) denotes a response of "never" What does \(P(\bar{N})\) represent, and what is its value?
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