Chapter 10

Express the sum in terms of summation notation. (Answers are not unique.) $$2-4+8-16+32-64$$

If the probability of a basketball player's making a free throw is \(0.9\), find the probability that the player makes at least 1 of 2 free throws.

A true-or-false test has 20 questions. (a) In how many different ways can the test be completed? (b) In how many different ways can a student answer 10 questions correctly?

Use the binomial theorem to expand and simplify. $$ \left(\frac{1}{x^{3}}-2 x\right)^{5} $$

Exer. 27-32: Find the smallest positive integer \(j\) for which the statement is true. Use the extended principle of mathematical induction to prove that the formula is true for every integer greater than \(j\). $$ n^{2}+18 \leq n^{3} $$

Book arrangement In how many different ways can ten books be arranged on a shelf?

Express the sum in terms of summation notation. (Answers are not unique.) $$\frac{1}{4}-\frac{1}{12}+\frac{1}{36}-\frac{1}{108}$$

The winner of the sevengame NBA championship series is the team that wins four games. In how many different ways can the series be extended to seven games?

Use the binomial theorem to expand and simplify. $$ \left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{5} $$

Exer. 27-32: Find the smallest positive integer \(j\) for which the statement is true. Use the extended principle of mathematical induction to prove that the formula is true for every integer greater than \(j\). $$ 5+\log _{2} n \leq n $$

Semaphore With six different flags, how many different signals can be sent by placing three flags, one above the other, on a flag pole?

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ 4+11+18+25+32 $$

Express the sum in terms of summation notation. (Answers are not unique.) $$3+\frac{3}{5}+\frac{3}{25}+\frac{3}{123}+\frac{3}{625}$$

A geometric design is determined by joining every pair of vertices of an octagon (see the figure). (a) How many triangles in the design have their three vertices on the octagon? (b) How many quadrilaterals in the design have their four vertices on the octagon?

Use the binomial theorem to expand and simplify. $$ \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{5} $$

Exer. 27-32: Find the smallest positive integer \(j\) for which the statement is true. Use the extended principle of mathematical induction to prove that the formula is true for every integer greater than \(j\). $$ n^{2} \leq 2^{n} $$

Selecting books In how many different ways can five books be selected from a twelve-volume set of books?

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ 3+8+13+18+23 $$

Find the sum of the infinite geometric series if it exists. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$$

An ice cream parlor stocks 31 different flavors and advertises that it serves almost 4500 different triple scoop cones, with each scoop being a different flavor. How was this number obtained?

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(3 c^{2 / 5}+c^{4 / 5}\right)^{25} ; \quad \text { first three terms } $$

Exer. 27-32: Find the smallest positive integer \(j\) for which the statement is true. Use the extended principle of mathematical induction to prove that the formula is true for every integer greater than \(j\). $$ 2 n+2 \leq 2^{n} $$

Radio call letters How many four-letter radio station call letters can be formed if the first letter must be \(\mathrm{K}\) or \(\mathrm{W}\) and repetitions (a) are not allowed? (b) are allowed?

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ 4+11+18+\cdots+466 $$

Find the sum of the infinite geometric series if it exists. $$2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots$$

A fast food restaurant advertises that it offers any combination of 8 condiments on a hamburger, thus giving a customer 256 choices. How was this number obtained?

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(x^{3}+5 x^{-2}\right)^{20}, \quad \text { first three terms } $$

Fraternity designations There are 24 letters in the Greek alphabet. How many fraternities may be specified by choosing three Greek letters if repetitions (a) are not allowed? (b) are allowed?

Exer. 27-32: Find the smallest positive integer \(j\) for which the statement is true. Use the extended principle of mathematical induction to prove that the formula is true for every integer greater than \(j\). $$ n \log _{2} n+20 \leq n^{2} $$

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ 3+8+13+\cdots+463 $$

Find the sum of the infinite geometric series if it exists. $$1.5+0.015+0.00015+\cdots$$

True-or-false test A true-or-false test consists of eight questions. If a student guesses the answer for each question, find the probability that (a) eight answers are correct (b) seven answers are correct and one is incorrect (c) six answers are correct and two are incorrect (d) at least \(\operatorname{six}\) answers are correct

A committee is going to select 30 students from a pool of 1000 to receive scholarships. How may ways could the students be selected if each scholarship is worth (a) the same amount? (b) a different amount?

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(4 z^{-1}-3 z\right)^{15} ; \quad \text { last three terms } $$

Phone numbers How many ten-digit phone numbers can be formed from the digits \(0,1,2,3, \ldots, 9\) if the first digit may not be 0 ?

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ \frac{3}{7}+\frac{6}{11}+\frac{9}{15}+\frac{12}{19}+\frac{15}{23}+\frac{18}{27} $$

\(\sum_{k=1}^{5}(2 k-7) \quad\)

Find the sum of the infinite geometric series if it exists. $$1-0.1+0.01-0.001+\cdots$$

Committee selection A 6-member committee is to be chosen by drawing names of individuals from a hat. If the hat contains the names of 8 men and 14 women, find the probability that the committee will consist of 3 men and 3 women.

Twelve sprinters are running a heat; those with the best four times will advance to the finals. (a) In how many ways can this group of four be selected? (b) If the four best times will be seeded (ranked) in the finals, in how many ways can this group of four be selected and seeded?

Without expanding completely, find the indicated term(s) in the expansion of the expression. \(\left(s-2 t^{3}\right)^{12}\) last three terms

Express the sum in terms of \(n\). $$ \sum_{k=1}^{n}\left(3 k^{2}-2 k+1\right) $$

Baseball batting order After selecting nine players for a baseball game, the manager of the team arranges the batting order so that the pitcher bats last and the best hitter bats third. In how many different ways can the remainder of the batting order be arranged?

Exer. 29-34: Express the sum in terms of summation notation. (Answers are not unique.) $$ \frac{5}{13}+\frac{10}{11}+\frac{15}{9}+\frac{20}{7} $$

\(\sum_{k=1}^{6}(10-3 k)\)

Find the sum of the infinite geometric series if it exists. $$\sqrt{2}-2+\sqrt{8}-4+\cdots$$

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(\frac{3}{c}+\frac{c^{2}}{4}\right)^{7} ; \quad \text { sixth term } $$

Express the sum in terms of \(n\). $$ \sum_{k=1}^{n}(2 k-3)^{2} $$

ATM access code A customer remembers that \(2,4,7\), and 9 are the digits of a four-digit access code for an automatic bank-teller machine. Unfortunately, the customer has forgotten the order of the digits. Find the largest possible number of trials necessary to obtain the correct code.

Exer. 35-36: Express the sum in terms of summation notation and find the sum. $$ 8+19+30+\cdots+16,805 $$