Chapter 8

In Exercises 39-48, find the term indicated in each expansion. $$\left(x-\frac{1}{2}\right)^{y} ; \text { fourth term }$$

Express each repeating decimal as a fraction in lowest terms. $$0.5=\frac{5}{10}+\frac{5}{100}+\frac{5}{1000}+\frac{5}{10,000}+\dots$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$2+2^{2}+2^{3}+\dots+2^{11}$$

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) In a production of West Side Story, eight actors are considered for the male roles of Tony, Riff, and Bernardo. In how many ways can the director cast the male roles?

Exercises \(46-48\) will help you prepare for the material covered in ehe next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). $$\begin{array}{l} (a+b)^{1}=a+b \\ (a+b)^{2}=a^{2}+2 a b+b^{2} \\ (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\ (a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ (a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5} \end{array}$$ Describe the pattern for the exponents on \(a\).

The mathematics department of a college has 8 male professors, 11 female professors, 14 male teaching assistants, and 7 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a professor or a female.

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{20}(6 i-4)$$

In Exercises 39-48, find the term indicated in each expansion. $$\left(x+\frac{1}{2}\right)^{8} ; \text { fourth term }$$

Express each repeating decimal as a fraction in lowest terms. $$0 . \overline{1}=\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10,000}+\cdots$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$5+5^{2}+5^{3}+\dots+5^{12}$$

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) Nine bands have volunteered to perform at a benefit concert, but there is only enough time for five of the bands to play. How many lineups are possible?

Exercises \(46-48\) will help you prepare for the material covered in ehe next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). $$\begin{array}{l} (a+b)^{1}=a+b \\ (a+b)^{2}=a^{2}+2 a b+b^{2} \\ (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\ (a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ (a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5} \end{array}$$ Describe the pattern for the exponents on \(b\).

A single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{30}(-3 i+5)$$

In Exercises 39-48, find the term indicated in each expansion. $$\left(x^{2}+y\right)^{22} ;$$ the term containing \(y^{14}\)

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{47}=\frac{47}{100}+\frac{47}{10,000}+\frac{47}{1,000,000}+\dots $$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1+2+3+\dots+30$$

Exercises \(46-48\) will help you prepare for the material covered in ehe next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). $$\begin{array}{l} (a+b)^{1}=a+b \\ (a+b)^{2}=a^{2}+2 a b+b^{2} \\ (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\ (a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ (a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5} \end{array}$$ Describe the pattern for the sum of the exponents on the variables in each term.

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) How many arrangements can be made using four of the letters of the word COMBINE if no letter is to be used more than once?

A single die is rolled twice. Find the probability of rolling a 5 the first time and a 1 the second time.

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{40}(-2 i+6)$$

In Exercises 39-48, find the term indicated in each expansion. $$(x+2 y)^{10} ;$$ the term containing \(y^{6}\)

Express each repeating decimal as a fraction in lowest terms. $$0 . \overline{83}=\frac{83}{100}+\frac{83}{10,000}+\frac{83}{1,000,000}+\dots$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1+2+3+\dots+40$$

Use the formula for \(_{n} C_{r}\) to solve An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

A single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{10} 4 i$$

In Exercises 49-52, use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{3}+x^{-2}\right)^{4}$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{14}{14+1}$$

Use the formula for \(_{n} C_{r}\) to solve A four-person committee is to be elected from an organization's membership of 11 people. How many different committees are possible?

A single die is rolled twice. Find the probability of rolling an odd number the first time and a number less than 3 the second time.

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{50}(-4 i)$$

In Exercises 49-52, use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{2}+x^{-3}\right)^{4}$$

Express each repeating decimal as a fraction in lowest terms. $$0.529$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$\frac{1}{3}+\frac{2}{4}+\frac{3}{5}+\dots+\frac{16}{16+2}$$

Use the formula for \(_{n} C_{r}\) to solve Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

If you toss a fair coin six times, what is the probability of getting all heads?

In Exercises 49-52, use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{\frac{1}{3}}-x^{-\frac{1}{3}}\right)^{3}$$

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=n+5 $$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$4+\frac{4^{2}}{2}+\frac{4^{3}}{3}+\dots+\frac{4^{n}}{n}$$

Use the formula for \(_{n} C_{r}\) to solve There are 14 standbys who hope to get seats on a flight, but only 6 seats are available on the plane. How many different ways can the 6 people be selected?

If you toss a fair coin seven times, what is the probability of getting all tails?

In Exercises 49-52, use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{\frac{2}{3}}-\frac{1}{\sqrt[3]{x}}\right)^{3}$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$-\frac{1}{9}+\frac{2}{9^{2}}+\frac{3}{9^{3}}+\dots+\frac{n}{9^{n}}$$

Use the formula for \(_{n} C_{r}\) to solve You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?

The probability that South Florida will be hit by a major hurricane (category 4 or 5 ) in any single year is \(\frac{1}{16}\) a. What is the probability that South Florida will be hit by a major hurricane two years in a row? b. What is the probability that South Florida will be hit by a major hurricane in three consecutive years? c. What is the probability that South Florida will not be hit by a major hurricane in the next ten years? d. What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years?

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1+3+5+\dots+(2 n-1)$$

Use the formula for \(_{n} C_{r}\) to solve Of the 100 people in the U.S. Senate, 18 serve on the Foreign Relations Committee. How many ways are there to select Senate members for this committee (assuming party affiliation is not a factor in selection)?

Describe the difference between theoretical probability and empirical probability.

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$a+a r+a r^{2}+\dots+a r^{n-1}$$