Chapter 9

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling any sum other than 5 or 6 .

For the following exercises, use the Binomial Theorem to expand the binomial \(f(x)=(x+3)^{4}\). Then find and graph each indicated sum on one set of axes. Find and graph \(f_{3}(x),\) such that \(f_{3}(x)\) is the sum of the first three terms of the expansion.

In horse racing, a "trifecta" occurs when a bettor wins by selecting the first three finishers in the exact order (1st place, 2nd place, and 3rd place). How many different trifectas are possible if there are 14 horses in a race?

For the following exercises, find the sum of the infinite geometric series. \(4+2+1+\frac{1}{2} \ldots\)

For the following exercises, find the specified term for the geometric sequence given. Let \(a_{1}=4, a_{n}=-3 a_{n-1} .\) Find \(a_{8}\)

For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence. \(a_{n}=24-4 n\) \(a_{n}=\frac{1}{2} n-\frac{1}{2}\)

For the following exercises, evaluate the factorial. $$ \frac{100 !}{99 !} $$

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: A head on the coin or a club

For the following exercises, use the Binomial Theorem to expand the binomial \(f(x)=(x+3)^{4}\). Then find and graph each indicated sum on one set of axes. Find and graph \(f_{4}(x),\) such that \(f_{4}(x)\) is the sum of the first four terms of the expansion.

A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different Tshirts are there to choose from?

For the following exercises, find the sum of the infinite geometric series. \(-1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64} \ldots\)

For the following exercises, find the specified term for the geometric sequence given. Let \(a_{n}=-\left(-\frac{1}{3}\right)^{n-1} \cdot\) Find \(a_{12}\).

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{3,5,7, \ldots\\} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n !}{n^{2}} $$

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: A tail on the coin or red ace

Hector wants to place billboard advertisements throughout the county for his new business. How many ways can Hector choose 15 neighborhoods to advertise in if there are 30 neighborhoods in the county?

For the following exercises, find the sum of the infinite geometric series. \(\sum_{\infty}^{k=1} 3 \cdot\left(\frac{1}{4}\right)^{k-1}\)

For the following exercises, find the number of terms in the given finite geometric sequence. \(a_{n}=\\{-1,3,-9, \ldots, 2187\\}\)

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{32,24,16, \ldots\\} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{3 \cdot n !}{4 \cdot n !} $$

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: A head on the coin or a face card

In the expansion of \((5 x+3 y)^{n},\) each term has the form \(\left(\begin{array}{l}n \\ k\end{array}\right) a^{n-k} b^{k},\) where \(k\) successively takes on the value \(0,1,2, \ldots, n .\) If \(\left(\begin{array}{l}n \\ k\end{array}\right)=\left(\begin{array}{l}7 \\\ 2\end{array}\right),\) what is the corresponding term?

An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?

For the following exercises, find the sum of the infinite geometric series. \(\sum_{n=1}^{\infty} 4.6 \cdot 0.5^{n-1}\)

For the following exercises, find the number of terms in the given finite geometric sequence. \(a_{n}=\left\\{2,1, \frac{1}{2}, \ldots, \frac{1}{1024}\right\\}\)

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{-5,95,195, \ldots\\} $$

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: No aces

In the expansion of \((a+b)^{n}\), the coefficient of \(a^{n-k} b^{k}\) is the same as the coefficient of which other term?

How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of 8 freshmen and 11 juniors?

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \(\$ 50 ;\) total deposits: 60 ; interest rate: \(5 \%\), compounded monthly

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{-17,-217,-417, \ldots\\} $$

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{100 \cdot n}{n(n-1) !} $$

For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag, a person grabs 5 M\&Ms. What is the probability of getting all blue M\&Ms?

Consider the expansion of \((x+b)^{40} .\) What is the exponent of \(b\) in the \(k\) the term?

How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \(\$ 150 ;\) total deposits: 24 ; interest rate: \(3 \%\), compounded monthly

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{1.8,3.6,5.4, \ldots\\} $$

For the following exercises, graph the first five terms of the indicated sequence \(a_{n}=\frac{(-1)^{n}}{n}+n\)

For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag, a person grabs 5 M\&Ms. What is the probability of getting 4 blue M\&Ms?

Find \(\left(\begin{array}{c}n \\\ k-1\end{array}\right)+\left(\begin{array}{l}n \\ k\end{array}\right)\) and write the answer as a binomial coefficient in the form \(\left(\begin{array}{l}n \\ k\end{array}\right)\). Prove it.

A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestra's 10 cellists and 16 violinists in order of musical proficiency. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \(\$ 450 ;\) total deposits: \(60 ;\) interest rate: \(4.5 \%\), compounded quarterly

For the following exercises, use the information provided to graph the first five terms of the geometric sequence. \(a_{1}=1, \quad r=\frac{1}{2}\)

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{-18.1,-16.2,-14.3, \ldots\\} $$

For the following exercises, graph the first five terms of the indicated sequence \(a_{n}=\left\\{\begin{array}{ll}\frac{4+n}{2 n} & \text { if } n \text { is even } \\ 3+n & \text { if } n \quad \text { is odd }\end{array}\right.\)

For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag, a person grabs 5 M\&Ms. What is the probability of getting 3 blue M\&Ms?

Which expression cannot be expanded using the Binomial Theorem? Explain. $$\begin{array}{l} \left(x^{2}-2 x+1\right) \\ (\sqrt{a}+4 \sqrt{a}-5)^{8} \\ \left(x^{3}+2 y^{2}-z\right)^{5} \\ \left(3 x^{2}-\sqrt{2 y^{3}}\right)^{12} \end{array}$$

A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers-different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?

For the following exercises, use the information provided to graph the first five terms of the geometric sequence. \(a_{1}=3, \quad a_{n}=2 a_{n-1}\)

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{15.8,18.5,21.2, \ldots\\} $$