Chapter 9

Find the indicated \(n\) th partial sum of the arithmetic sequence. $$2,8,14,20, \ldots ; n=25$$

Three points that are not collinear determine three lines. How many lines are determined by nine points, of which no three are collinear?

Using Summation Notation Use summation notation to write the sum. $$7+14+28+\cdots+896$$

Find the coefficient \(a\) of the given term in the expansion of the binomial. Binomial =\((4 x-y)^{10}\) Term = \(a x^{2} y^{8}\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{(-1)^{2 n}}{(2 n) !}$$

Find the indicated \(n\) th partial sum of the arithmetic sequence. $$7.2,6.4,5.6,4.8, . . . . ; n=10$$

Powerball is played with 59 white balls, numbered 1 through \(59,\) and 35 red balls, numbered 1 through \(35 .\) Five white balls and one red ball, the Powerball, are drawn. In how many ways can a player select the six numbers?

A sales representative makes sales on approximately one-fifth of all calls. On a given day, the representative contacts six potential clients. What is the probability that a sale will be made with (a) all six contacts, (b) none of the contacts, and (c) at least one contact?

Using Summation Notation Use summation notation to write the sum. $$2-\frac{1}{2}+\frac{1}{8}-\cdots+\frac{1}{2048}$$

Find the coefficient \(a\) of the given term in the expansion of the binomial. Binomial =\((3 x-2 y)^{9}\) Term = \(a x^{6} y^{3}\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 0.) $$a_{n}=\frac{(-1)^{2 n+1}}{(2 n+1) !}$$

Find the indicated \(n\) th partial sum of the arithmetic sequence. $$4.2,3.7,3.2,2.7, . . . ; n=12$$

A law office interviews paralegals for 10 openings. There are 13 paralegals with two years of experience and 20 paralegals with one year of experience. How many combinations of seven paralegals with two years of experience and three paralegals with one year of experience are possible?

Assume that the probability of the birth of a child of a particular sex is \(50 \% .\) In a family with four children, what is the probability that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy?

Simplify the factorial expression. $$\frac{2 !}{4 !}$$

Find the indicated \(n\) th partial sum of the arithmetic sequence. $$a_{1}=100, a_{25}=220, n=25$$

A six-member research committee is to be formed having one administrator, three faculty members, and two students. There are seven administrators, 12 faculty members, and 25 students in contention for the committee. How many six-member committees are possible?

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 10\left(\frac{4}{5}\right)^{n}$$

Find the coefficient \(a\) of the given term in the expansion of the binomial. Binomial = \(\left(x^{2}+y\right)^{10}\) Term = \(a x^{8} y^{6}\)

Simplify the factorial expression. $$\frac{5 !}{7 !}$$

Find the indicated \(n\) th partial sum of the arithmetic sequence. $$a_{1}=15, a_{100}=307, n=100$$

Sociology The number of possible interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships that are possible in groups of people of sizes (a) \(3,(b) 8,(c) 12,\) and (d) \(20\).

You and a friend agree to meet at your favorite fast food restaurant between 5: 00 and 6: 00 P.M. The one who arrives first will wait 15 minutes for the other, after which the first person will leave (see figure). What is the probability that the two of you will actually meet, assuming that your arrival times are random within the hour?

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 6\left(\frac{2}{3}\right)^{n}$$

Find the coefficient \(a\) of the given term in the expansion of the binomial. Binomial = \(\left(z^{2}+y\right)^{12}\) Term = \(a z^{14} y^{5}\)

Simplify the factorial expression. $$\frac{12 !}{4 ! \cdot 8 !}$$

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{50} n$$

You are dealt five cards from an ordinary deck of 52 playing cards. In how many ways can you get a full house? (A full house consists of three of one kind and two of another. For example, \(8-8-8-5-5\) and \(\mathrm{K}-\mathrm{K}-\mathrm{K}-10-10\) are full houses.)

Determine whether the statement is true or false. Justify your answer. If the probability of an outcome in a sample space is 1 then the probability of the other outcomes in the sample space is \(0 .\)

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 5\left(-\frac{1}{2}\right)^{n}$$

Use Pascal's Triangle to find the binomial coefficient. \(_{7} C_{4}\)

Simplify the factorial expression. $$\frac{10 !}{5 ! \cdot 3 !}$$

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{100} 2 n$$

A shipment of 30 flat screen televisions contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at least two good units?

Determine whether the statement is true or false. Justify your answer. If \(A\) and \(B\) are independent events with nonzero probabilities, then \(A\) can occur when \(B\) occurs.

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 5\left(-\frac{1}{4}\right)^{n}$$

Use Pascal's Triangle to find the binomial coefficient. \(_{6} C_{3}\)

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{30} n-\sum_{n=1}^{10} n$$

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of a polygon.) Pentagon

Consider a group of \(n\) people. (a) Explain why the following pattern gives the probability that the \(n\) people have distinct birthdays. $$\begin{array}{l} n=2: \frac{365}{365} \cdot \frac{364}{365}=\frac{365 \cdot 364}{365^{2}} \\ n=3: \frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365}=\frac{365 \cdot 364 \cdot 363}{365^{3}} \end{array}$$ (b) Use the pattern in part (a) to write an expression for the probability that four people \((n=4)\) have distinct birthdays. (c) Let \(P_{n}\) be the probability that the \(n\) people have distinct birthdays. Verify that this probability can be obtained recursively by $$P_{1}=1 \quad \text { and } \quad P_{n}=\frac{365-(n-1)}{365} P_{n-1}$$ (d) Explain why \(Q_{n}=1-P_{n}\) gives the probability that at least two people in a group of \(n\) people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline n & 10 & 15 & 20 & 23 & 30 & 40 & 50 \\ \hline P_{n} & & & & & & & \\ \hline Q_{n} & & & & & & & \\ \hline \end{array}$$ (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than \(\frac{1}{2} ?\) Explain.

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=1}^{\infty} 2\left(\frac{7}{3}\right)^{n-1}$$

Use Pascal's Triangle to find the binomial coefficient. \(_{6} C_{5}\)

Simplify the factorial expression. $$\frac{(n+3) !}{n !}$$

Find the partial sum without using a graphing utility. $$\sum_{n=51}^{100} n-\sum_{n=1}^{50} n$$

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of a polygon.) Hexagon

Write a paragraph describing in your own words the difference between mutually exclusive events and independent events.

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=1}^{\infty} 8\left(\frac{5}{3}\right)^{n-1}$$

Use Pascal's Triangle to find the binomial coefficient. \(_{5} C_{2}\)

Simplify the factorial expression. $$\frac{(n+2) !}{n !}$$

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{500}(n+8)$$