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A study was dont to determine the efficacy of three different drugs-A, \(B\), and \(C\)-in relieving headache pain. Over the period covered by the study, 50 subjects were given the chance to use all three drugs. The following results were obtained: 21 reported relief from drug \(A\). 21 reported relief from drug \(B\). 31 reported relief from drug \(C\). 9 reported relief from both drugs \(A\) and \(B\). 14 reported relief from both drugs \(A\) and \(C\). 15 reported relief from both drugs \(B\) and \(C\). 41 reported relicf from at least one of the drugs. Note that some of the 21 subjects who reported relief from drug \(A\) may also have reported relief from drugs \(B\) or \(C . A\) similar occurrence may be true for the other data. a. How many people got relief from none of the drugs? b. How many people got relief from all three drugs? c. Let \(A\) be the set of all sabjects who got relief from drug \(A, B\) the set of all subjects who got relief from drug \(B\), and \(C\) the set of all subjects who got relief from drug \(C\). Fill in the numbers for all eight regions of the diagram below. d. How many subjects got relief from \(A\) only?

A coin is loaded so that the probability of heads is \(0.7\) and the probability of tails is \(0.3 .\) Suppose that the coin is tossed ten times and that the results of the tosses are mutually independent. a. What is the probability of obtaining exactly seven heads? b. What is the probability of obtaining exactly ten heads? c. What is the probability of obtaining no heads? d. What is the probability of obtaining at least one head?

An interesting use of the inclusion/exclusion rule is to check survey numbers for consistency. For example, suppose a public opinion polltaker reports that out of a national sample of 1,200 adults, 675 are married, 682 are from 20 to 30 years old, 684 are female, 195 are married and are from 20 to 30 years old, 467 are married females, 318 are females from 20 to 30 years old, and 165 are married females from 20 to 30 years old. Are the polltaker's figures consistent? Could they have occurred as a result of an actual sample survey?

If the largest of 56 consecutive integers is 279 , what is the smallest?

a. How many ways can the letters of the word \(A L G O R I T H M\) be arranged in a row? b. How many ways can the letters of the word \(A L G O R I T H M\) be arranged in a row if \(A\) and \(L\) must remain together (in order) as a unit? c. How many ways can the letters of the word \(A L G O R I T H M\) be arranged in a row if the letters GOR must remain together (in order) as a unit?

Suppose that ten items are chosen at random from a large batch delivered to a company. The manufacturer claims that just \(3 \%\) of the items in the batch are defective. Assume that the batch is large enough so that even though the selection is made without replacement, the number \(0.03\) can be used to approximate the probability that any one of the ten items is defective. In addition, assume that because the items are chosen at random, the outcomes of the choices are mutually independent. Finally, assume that the manufacturer's claim is correct. a. What is the probability that none of the ten is defective? b. What is the probability that at least one of the ten is defective? c. What is the probability that exactly four of the ten are defective? d. What is the probability that at most two of the ten are defective?

How many positive integers less than 1,000 have no common factors with 1,000 ?

Six people attend the theater together and sit in a row with exactly six seats. a. How many ways can they be seated together in the row? b. Suppose one of the six is a doctor who must sit on the aisle in case she is paged. How many ways can the people be seated together in the row with the doctor in an aisle seat? c. Suppose the six people consist of three married couples and each couple wants to sit together with the husband on the left. How many ways can the six be seated together in the row?

Express each of the sums in \(24-35\) in closed form (without using a summation symbol and without using an ellipsis \(\cdots\) ). $$ \sum_{i=0}^{m}\left(\begin{array}{c} m \\ i \end{array}\right) p^{m-i} q^{2 i} $$

How many permutations of \(a b c d e\) are there in which the first character is \(a, b\), or \(c\) and the last character is \(c, d\), or \(e\) ?