Chapter 9

Evaluate the factorial. $$\left(\frac{12}{6}\right) !$$

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

Use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. $$ \sum_{n=1}^{10}-2 \cdot\left(\frac{1}{2}\right)^{n-1} $$

A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?

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 \(f_{2}(x)\), such that \(f_{2}(x)\) is the sum of the fi st two terms of the expansion.

For the following exercises, write an explicit formula for each geometric sequence. $$ a_{n}=\left\\{3,-1, \frac{1}{3},-\frac{1}{9}, \ldots\right\\} $$

For the following exercises, evaluate the factorial. $$ \frac{12 !}{6 !} $$

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. $$ a_{n}=24-4 n $$

Evaluate the factorial. $$\frac{12 !}{6 !}$$

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.

Find the sum of the infinite geometric series. $$ 4+2+1+\frac{1}{2} \ldots $$

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

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 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 fi st three terms of the expansion.

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

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. $$ a_{n}=\frac{1}{2} n-\frac{1}{2} $$

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

Find the sum of the infinite geometric series. $$ -1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64} \ldots $$

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

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 fi st four terms of the expansion.

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, find the specified term for the geometric sequence given. $$a_{n}=-\left(-\frac{1}{3}\right)^{n-1} . \text { Find } a_{12}$$

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

Find the sum of the infinite geometric series. $$ \sum_{\infty}^{k=1} 3 \cdot\left(\frac{1}{4}\right)^{k-1} $$

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 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. $$ a_{n}=\frac{3 \cdot n !}{4 \cdot n !} $$

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

Find the sum of the infinite geometric series. $$ \sum_{n=1}^{\infty} 4.6 \cdot 0.5^{n-1} $$

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 number of terms in the given finite geometric sequence. $$ a_{n}=\left\\{2,1, \frac{1}{2}, \ldots, \frac{1}{1024}\right\\} $$

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 \(\left(\begin{array}{c}n \\ k\end{array}\right)=\left(\begin{array}{l}7 \\\ 2\end{array}\right),\) what is the corresponding term?

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

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n !}{n^{2}-n-1} $$

Write the first four terms of the sequence. $$a_{n}=\frac{n !}{n^{2}-n-1}$$

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

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

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

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

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. $$ a_{n}=\frac{100 \cdot n}{n(n-1) !} $$

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?

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?

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

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