Chapter 8

Write the sum of each geometric series as a rational number. $$0.8+0.08+0.008+0.0008+\cdots$$

Write the indicated term of each binomial expansion. Fifteenth term of \(\left(a^{2}+b\right)^{22}\).

Use the fundamental principle of counting or permutations to solve each problem. License Plates For many years, the state of California used 3 letters followed by 3 digits on its automobile license plates. (a) How many different license plates are possible with this arrangement? (b) When the state ran out of new plates, the order was reversed to 3 digits followed by 3 letters. How many additional plates were then possible? (c) Several years ago, the plates described in part (b) were also used up. The state then issued plates with 1 letter, followed by 3 digits, and then 3 letters. How many plates does this scheme provide?

Find the sum for each series. $$\sum_{i=1}^{5}\left(i^{2}-2 i\right)$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{31}=5580, a_{31}=360$$

A die is rolled 12 times. Approximate the probability of rolling the following. Exactly 6 ones

Write the sum of each geometric series as a rational number. $$0.7+0.07+0.007+0.0007+\cdots$$

Write the indicated term of each binomial expansion. Twelfth term of \(\left(2 x+y^{2}\right)^{16}\).

Use the fundamental principle of counting or permutations to solve each problem. Telephone Numbers How many 7-digit telephone numbers are possible if the first digit cannot be 0 and (a) only odd digits may be used? (b) the telephone number must be a multiple of 10 (that is, it must end in 0 )? (c) the telephone number must be a multiple of \(100 ?\) (d) the first 3 digits are \(481 ?\) (e) no repetitions are allowed?

Find the sum for each series. $$\sum_{i=3}^{6}\left(2 i^{2}+1\right)$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{12}=-108, a_{12}=-19$$

A die is rolled 12 times. Approximate the probability of rolling the following. No more than 3 ones

Write the sum of each geometric series as a rational number. $$0.45+0.0045+0.000045+\cdots$$

Write the indicated term of each binomial expansion. Fifteenth term of \(\left(x-y^{3}\right)^{20}\).

Use the fundamental principle of counting or permutations to solve each problem. Seating People in a Row In an experiment on social interaction, 6 people will sit in 6 seats in a row. In how many ways can this be done?

Find the sum for each series. $$\sum_{i=1}^{5}\left(3^{i}-4\right)$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{25}=650, a_{25}=62$$

A die is rolled 12 times. Approximate the probability of rolling the following. No more than 1 one

Write the sum of each geometric series as a rational number. $$0.36+0.0036+0.000036+\cdots$$

Write the indicated term of each binomial expansion. Tenth term of \(\left(a^{3}+3 b\right)^{11}\).

Use the fundamental principle of counting or permutations to solve each problem. Chemistry Experiment In how many ways can 7 of 10 chemicals be added to a beaker for an experiment?

Find the sum for each series. $$\sum_{i=1}^{4}\left[(-2)^{i}-3\right]$$

The table gives the results of a survey of \(282,549\) freshmen from a recent class year at 437 of the nation's baccalaureate colleges and universities. $$\begin{array}{|lc|c|c|c|} \hline \begin{array}{l} \text { Number of Colleges } \\ \text { Applied to } \end{array} & 1 & 2 \text { or } 3 & 4-6 & 7 \text { or more } \\ \hline \begin{array}{l} \text { Percent (as a } \\ \text { decimal) }The student applied to fewer than 4 colleges. \end{array} & 0.20 & 0.29 & 0.37 & 0.14 \end{array}$$$$\begin{aligned} &\text {Using the percents as probabilities, find the probability of}\\\ &\text { each event for a randomly selected student.} \end{aligned}$$

Write the sum of each geometric series as a rational number. $$0.378+0.000378+0.000000378+\cdots$$

Find the middle term of \(\left(3 x^{7}+2 y^{3}\right)^{8}\).

Use the fundamental principle of counting or permutations to solve each problem. Course Schedule Arrangement \(\quad\) A business school offers courses in keyboarding, spreadsheets, transcription, business English, technical writing, and accounting. In how many ways can a student arrange a schedule if 3 courses are taken?

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\). $$\sum_{i=1}^{5} x_{i}$$

The table gives the results of a survey of \(282,549\) freshmen from a recent class year at 437 of the nation's baccalaureate colleges and universities. $$\begin{array}{|lc|c|c|c|} \hline \begin{array}{l} \text { Number of Colleges } \\ \text { Applied to } \end{array} & 1 & 2 \text { or } 3 & 4-6 & 7 \text { or more } \\ \hline \begin{array}{l} \text { Percent (as a } \\ \text { decimal) }The student applied to fewer than 4 colleges. \end{array} & 0.20 & 0.29 & 0.37 & 0.14 \end{array}$$$$\begin{aligned} &\text {Using the percents as probabilities, find the probability of}\\\ &\text { each event for a randomly selected student.} \end{aligned}$$The student applied to at least 2 colleges.

Write the sum of each geometric series as a rational number. $$0.297+0.000297+0.000000297+\cdots$$

Find the two middle terms of \(\left(-2 m^{-1}+3 n^{-2}\right)^{11}\).

Use the fundamental principle of counting or permutations to solve each problem. Course Schedule Arrangement If your college offers 400 courses, 20 of which are in mathematics, and your counselor arranges your schedule of 4 courses by random selection, how many schedules are possible that do not include a math course?

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\). $$\sum_{i=1}^{5}-x_{i}$$

The table gives the results of a survey of \(282,549\) freshmen from a recent class year at 437 of the nation's baccalaureate colleges and universities. $$\begin{array}{|lc|c|c|c|} \hline \begin{array}{l} \text { Number of Colleges } \\ \text { Applied to } \end{array} & 1 & 2 \text { or } 3 & 4-6 & 7 \text { or more } \\ \hline \begin{array}{l} \text { Percent (as a } \\ \text { decimal) }The student applied to fewer than 4 colleges. \end{array} & 0.20 & 0.29 & 0.37 & 0.14 \end{array}$$$$\begin{aligned} &\text {Using the percents as probabilities, find the probability of}\\\ &\text { each event for a randomly selected student.} \end{aligned}$$The student applied to more than 3 colleges.

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$12,24,48,96, \dots$$

Find the value of \(n\) for which the coefficients of the fifth and eighth terms in the expansion of \((x+y)^{n}\) are the same.

Use the fundamental principle of counting or permutations to solve each problem. Club Officer Choices In a club with 15 members, how many ways can a slate of 3 officers consisting of president, vice-president, and secretary/treasurer be chosen?

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\). $$\sum_{i=1}^{5}\left(2 x_{i}+3\right)$$

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$2,-10,50,-250, \dots$$

Find the term in the expansion of \((3+\sqrt{x})^{11}\) that contains \(x^{4}\).

Use the fundamental principle of counting or permutations to solve each problem. Batting Orders \(\quad\) A baseball team has 20 players. How many 9-player batting orders are possible?

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\). $$\sum_{i=1}^{4}\left(4-6 x_{i}\right)$$

The table gives the results of a survey of \(282,549\) freshmen from a recent class year at 437 of the nation's baccalaureate colleges and universities. $$\begin{array}{|lc|c|c|c|} \hline \begin{array}{l} \text { Number of Colleges } \\ \text { Applied to } \end{array} & 1 & 2 \text { or } 3 & 4-6 & 7 \text { or more } \\ \hline \begin{array}{l} \text { Percent (as a } \\ \text { decimal) }The student applied to fewer than 4 colleges. \end{array} & 0.20 & 0.29 & 0.37 & 0.14 \end{array}$$$$\begin{aligned} &\text {Using the percents as probabilities, find the probability of}\\\ &\text { each event for a randomly selected student.} \end{aligned}$$ The student applied to no colleges.

The probability that a male will be color blind is \(0.042 .\) Approximate the probabilities that in a group of 53 men, the following are true. (a) Exactly 5 are color blind. (b) No more than 5 are color blind. (c) At least 1 is color blind.

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$-48,-24,-12,-6, \dots$$

The factorial of a positive integer \(n\) can be computed as a product: \(n !=1 \cdot 2 \cdot 3 \cdot \cdots \cdot n\) Calculators and computers can evaluate factorials quickly. Before the days of technology, mathematicians developed a formula, called Stirling's formula, for approximating large factorials. Interestingly enough, it involves the irrational numbers \(\pi\) and \(e\). $$n ! \approx \sqrt{2 \pi n} \cdot n^{n} \cdot e^{-n}$$ As an example, the exact value of \(5 !\) is \(120,\) and Stirling's formula gives the approximation as 118.019168 with a graphing calculator. This is "off" by less than \(2,\) an error of only \(1.65 \% .\) Use a calculator and Stirling's formula to find the exact value of \(10 !\) and its approximation.

Use the fundamental principle of counting or permutations to solve each problem. Basketball Positions In how many ways can 5 players be assigned to the 5 positions on a basketball team, assuming that any player can play any position? In how many ways can 10 players be assigned to the 5 positions?

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\). $$\sum_{i=1}^{3}\left(3 x_{i}-x_{i}^{2}\right)$$

(a) Find the probabilities of having \(0,1,2,\) or 3 boys in a family of 3 children. (b) Find the probabilities of having \(0,1,2,3,4,5,\) or 6 girls in a family of 6 children.

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$625,125,25,5, \dots$$

Use the fundamental principle of counting or permutations to solve each problem. Letter Arrangement How many ways can all the letters of the word ELTON be arranged?