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Chapter 2: Problem 26

A California study concluded that by following 7 simple health rules a man's life can be extended by 11 years on the average and a woman's life by 7 years. These 7 rules are as follows: no smoking, regular exercise, use alcohol moderately, get, 7 to 8 hours of sleep, maintain proper weight, eat. breakfast, and do not eat between meals. In how many ways can a person adopt five of these rules to follow (a) If the person presently violates all 7 rules? (b) If the person never drinks and always eats breakfast?

Short Answer

Expert verified
The number of ways a person can adopt five rules given that (a) the person presently violates all 7 rules is 21 ways, and (b) the person never drinks and always eats breakfast is 1 way.

Step by step solution

01

Identify the Problem Type

This is a combination problem. That is because the order in which the rules are followed does not matter. These are not permutations.
02

Calculate the Combinations for First Case

In the case that a person violates all 7 rules, the number of ways in which he or she can choose any 5 to follow is given by \(C(7,5)\). To calculate this, use the combination formula: \(C(n, k) = \dfrac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and ! denotes factorial. Plug in n=7 and k=5 to get \(C(7,5) = \dfrac{7!}{5!(7-5)!} = 21\).
03

Calculate the Combinations for the Second Case

For a person who never drinks and always eats breakfast, those two rules are already followed and therefore don't count in the choice. So essentially, we are choosing 5 out of the remaining 5 rules. This is calculated as \(C(5,5)\). Using the combination formula again: \(C(5,5) = \dfrac{5!}{5!(5-5)!} = 1\).

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Key Concepts

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

Combination Formula
In combinatorics, the combination formula is a way to find how many ways you can choose items from a group, where the order does not matter. For instance, if you have a group of 7 health rules, and you want to pick 5 to follow, you'd use this formula. This is different from permutations where order does matter.

The general formula for combinations is:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]- **n** is the total number of items.- **k** is the number of items to choose.- **!** (factorial) represents the product of all positive integers up to that number.

This formula helps you avoid manually listing all possible combinations, which can be time-consuming and prone to error.
Factorial
A factorial, denoted by the symbol **!**, is a way to multiply a series of descending natural numbers. It's vital in permutations and combinations because it calculates the number of ways to arrange or choose items.

For example, the factorial of 5, written as **5!**, is calculated as: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
  • The factor of zero, written as **0!**, is always 1, by definition.
  • Factorials increase rapidly, as each integer is multiplied by all previous ones.
Understanding factorials allows us to compute combinations efficiently using the combination formula.
Probability
Probability is a mathematical concept that measures the likelihood of an event occurring. In the context of choosing rules to follow from a set, probability helps determine how likely a person will adopt certain combinations of behaviors.
  • **Probability Formula:** \[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]
  • Probability values range from 0 to 1, where 0 means impossible and 1 means certain.
While the original problem does not directly ask for probability, understanding it gives context to how these combinations could affect a person's chances of improving their lifestyle.
Statistics
Statistics involves collecting, analyzing, and interpreting data. In health studies, like the one mentioned in the exercise, statistics can help understand trends and draw conclusions about lifestyle impacts on longevity.
  • Statistics helps identify the effects of following certain health rules through collective data.
  • By analyzing combinations and probabilities, statisticians can make predictions about health outcomes.
Utilizing statistical methods, you can appreciate how small choices, like following specific health rules, can collectively influence larger population trends.

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Most popular questions from this chapter

By comparing appropriate regions of Venn diagrams, verify that (a) \((A \cap B) \cup\left(A n B^{\prime}\right)=A\); (b) \(\mathrm{A}^{\prime} n\left(B^{\prime} \cup C\right)=\left(A^{\prime} \mathrm{n} B^{\prime}\right) \mathrm{u}\left(A^{\prime} n C\right)\)

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Before the distribution of certain statistical software every fourth compact disk (CD) is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rate for the 4 testing programs are. respectively, \(0.01,0.03,0.02,\) and 0.01 (a) What is the probability that a CD was tested and failed any test? (b) Given that a CD was tested, what is the probability that it failed program 2 or \(3 ?\) (c) In a sample of 100 , how many CDs would you expect to bo rejected? (d) Given a CD was defective, what is the probability that it, was tested?

The probability that a vehicle entering the Luray Caverns has Canadian license plates is \(0.12 ;\) the probability that it is a camper is 0.28 ; and the probability that it is a camper with Canadian license plates is \(0.09 .\) What is the probability that (a) a camper entering the Luray Caverns has Canadian license plates? (b) a vehicle with Canadian license plates entering the Luray Caverns is a camper? (c) a vehicle entering the Luray Caverns does not have Canadian plates or is not a camper?

In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the probability that (a) the student took mathematics or history; (b) the student did not take either of these subjects; (c) the student took history but not mathematics.
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