Chapter 11

Find the middle term in the expansion of \(\left(\frac{1}{x}-x^{2}\right)^{12}\)

express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$ 6+8+10+12+\cdots+32 $$

To win in the New York State lottery, one must correctly select 6 numbers from 59 numbers. The order in which the selection is made does not matter. How many different selections are possible?

Explaining the Concepts Give an example of an event whose probability must be determined empirically rather than theoretically.

In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ and $$\left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \ldots$$ Find \(a_{10}+b_{10}\)

The probability that a smoker suffers from depression is 0.28. If five smokers are randomly selected, the probability that three of them will suffer from depression is the third term of the binomial expansion of

express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$ a+a r+a r^{2}+\dots+a r^{12} $$

In a race in which six automobiles are entered and there are no ties, in how many ways can the first four finishers come in?

Explaining the Concepts Write a probability word problem whose answer is one of the following fractions: \(\frac{1}{6}\) or \(\frac{1}{4}\) or \(\frac{1}{3}\)

In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ and $$\left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \ldots$$ Find \(a_{11}+b_{11}\)

The probability that a person in the general population suffers from depression is 0.12. If five people from the general population are randomly selected, the probability that three of them will suffer from depression is the third term of the binomial expansion of

express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$ a+a r+a r^{2}+\dots+a r^{14} $$

A book club offers a choice of 8 books from a list of 40. In how many ways can a member make a selection?

Explaining the Concepts Explain how to find the probability of an event not occurring. Give an example.

In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ and $$\left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \ldots$$ Find the difference between the sum of the first 10 terms of \(\left\\{a_{n}\right\\}\) and the sum of the first 10 terms of \(\left\\{b_{n}\right\\}\)

Explain how to evaluate \(\left(\begin{array}{l}{n} \\ {r}\end{array}\right) .\) Provide an example with your explanation.

express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$ a+(a+d)+(a+2 d)+\cdots+(a+n d) $$

Use a system of two equations in two variables, \(a_{1}\) and \(d\) Write a formula for the general term (the \(n\) th term) of the arithmetic sequence whose second term, \(a_{2},\) is 4 and whose sixth term, \(a_{6},\) is 16

A medical researcher needs 6 people to test the effectiveness of an experimental drug. If 13 people have volunteered for the test, in how many ways can 6 people be selected?

Explaining the Concepts What are mutually exclusive events? Give an example of two events that are mutually exclusive.

In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ and $$\left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \ldots$$ Find the difference between the sum of the first 11 terms of \(\left\\{a_{n}\right\\}\) and the sum of the first 11 terms of \(\left\\{b_{n}\right\\}\)

Describe the pattern in the exponents on \(a\) in the expansion of \((a+b)^{n}\)

express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$ (a+d)+\left(a+d^{2}\right)+\dots+\left(a+d^{n}\right) $$

Use a system of two equations in two variables, \(a_{1}\) and \(d\) Write a formula for the general term (the \(n\) th term) of the arithmetic sequence whose third term, \(a_{3},\) is 7 and whose eighth term, \(a_{8},\) is 17

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000\), second prize is \(\$ 500,\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

Explaining the Concepts Explain how to find or probabilities with mutually exclusive events. Give an example.

In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ and $$\left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \ldots$$ Find the product of the sum of the first 6 terms of \(\left\\{a_{n}\right\\}\) and the sum of the infinite series containing all the terms of \(\left\\{c_{n}\right\\}\)

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010 . Involve developing arithmetic sequences that model the data. Percentage of College Graduates for Americans Ages 25 and Older (Graph cannot copy) In \(1990,18.4 \%\) of American women ages 25 and older had graduated from college. On average, this percentage has increased by approximately 0.6 each year. A. Write a formula for the \(n\) th term of the arithmetic sequence that models the percentage of American women ages 25 and older who had graduated from college \(n\) years after 1989 B. Use the model from part (a) to project the percentage of American women ages 25 and older who will be college graduates by 2019

From a club of 20 people, in how many ways can a group of three members be selected to attend a conference?

Describe the pattern in the exponents on \(b\) in the expansion of \((a+b)^{n}\)

Explaining the Concepts Give an example of two events that are not mutually exclusive.

In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ and $$\left\\{c_{n}\right\\}=-2,1,-\frac{1}{2}, \frac{1}{4}, \ldots$$ Find the product of the sum of the first 9 terms of \(\left\\{a_{n}\right\\}\) and the sum of the infinite series containing all the terms of \(\left\\{c_{n}\right\\}\)

What is true about the sum of the exponents on \(a\) and \(b\) in any term in the expansion of \((a+b)^{n} ?\)

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010 . Involve developing arithmetic sequences that model the data. Percentage of College Graduates for Americans Ages 25 and Older (Graph cannot copy) In \(1990,24.4 \%\) of American men ages 25 and older had graduated from college. On average, this percentage has increased by approximately 0.3 each year. A. Write a formula for the \(n\) th term of the arithmetic sequence that models the percentage of American men ages 25 and older who had graduated from college \(n\) years after 1989 B. Use the model from part (a) to project the percentage of American men ages 25 and older who will be college graduates by 2019

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?

Explaining the Concepts Explain how to find or probabilities with events that are not mutually exclusive. Give an example.

In Exercises \(63-64,\) find \(a_{2}\) and \(a_{3}\) for each geometric sequence. $$ 8, a_{2}, a_{3}, 27 $$

How do you determine how many terms there are in a binomial expansion?

Company A pays \(\$ 24,000\) yearly with raises of \(\$ 1600\) per year. Company \(\mathrm{B}\) pays \(\$ 28,000\) yearly with raises of \(\$ 1000\) per year. Which company will pay more in year \(10 ?\) How much more?

How many different four-letter passwords can be formed from the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F},\) and \(\mathrm{G}\) if no repetition of letters is allowed?

Explaining the Concepts Explain how to find and probabilities with independent events. Give an example.

In Exercises \(63-64,\) find \(a_{2}\) and \(a_{3}\) for each geometric sequence. $$ 2, a_{2}, a_{3},-54 $$

Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.

Company A pays \(\$ 23,000\) yearly with raises of \(\$ 1200\) per year. Company B pays \(\$ 26,000\) yearly with raises of \(\$ 800\) per year. Which company will pay more in year \(10 ?\) How much more?

Nine comedy acts will perform over two evenings. Five of the acts will perform on the first evening and the order in which the acts perform is important. How many ways can the schedule for the first evening be made?

Explaining the Concepts The president of a large company with \(10,000\) employees is considering mandatory cocaine testing for every employee. The test that would be used is \(90 \%\) accurate, meaning that it will detect \(90 \%\) of the cocaine users who are tested, and that \(90 \%\) of the nonusers will test negative. This also means that the test gives \(10 \%\) false positive. Suppose that \(1 \%\) of the employees actually use cocaine. Find the probability that someone who tests positive for cocaine use is, indeed, a user. Hint: Find the following probability fraction: the number of employees who test positive and are cocaine users the number of employees who test positive This fraction is given by $$ 90 \% \text { of } 1 \% \text { of } 10,000 $$ the number who test positive who actually use cocaine plus the number who test positive who do not use cocaine What does this probability indicate in terms of the percentage of employees who test positive who are not actually users? Discuss these numbers in terms of the issue of mandatory drug testing. Write a paper either in favor of or against mandatory drug testing, incorporating the actual percentage accuracy for such tests.

Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises \(65-68\) In Exercises \(65-66,\) suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the fifteenth day of the month?

Explain how to find a particular term in a binomial expansion without having to write out the entire expansion.

Using 15 flavors of ice cream, how many cones with three different flavors can you create if it is important to you which flavor goes on the top, middle, and bottom?

Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises \(65-68\) In Exercises \(65-66,\) suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the thirtieth day of the month?