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How many subsets does the set $\left\{1,\text{}3,\text{}5,\text{}\dots ,\text{}99\right\}$ have?
${2}^{50}=1.13\times {10}^{15}$
A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose from the following additional services: massage, body scrub, manicure, pedicure, facial, and straight-razor shave. How many ways are there to order additional services at the day spa?
How many distinct ways can the word DEADWOOD be arranged?
$\frac{8!}{3!2!}=3360$
How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?
Evaluate the binomial coefficient $\text{\hspace{0.17em}}\left(\begin{array}{c}23\\ 8\end{array}\right).$
$490\text{,}314$
Use the Binomial Theorem to expand ${\left(3x+\frac{1}{2}y\right)}^{6}.$
Use the Binomial Theorem to write the first three terms of ${\left(2a+b\right)}^{17}.$
$131\text{,}072{a}^{17}\text{+}1\text{,}114\text{,}112{a}^{16}b\text{+}4\text{,}456\text{,}448{a}^{15}{b}^{2}$
Find the fourth term of ${\left(3{a}^{2}-2b\right)}^{11}$ without fully expanding the binomial.
For the following exercises, assume two die are rolled.
Construct a table showing the sample space.
1 | 2 | 3 | 4 | 5 | 6 | |
1 | 1, 1 | 1, 2 | 1, 3 | 1, 4 | 1, 5 | 1, 6 |
2 | 2, 1 | 2, 2 | 2, 3 | 2, 4 | 2, 5 | 2, 6 |
3 | 3, 1 | 3, 2 | 3, 3 | 3, 4 | 3, 5 | 3, 6 |
4 | 4, 1 | 4, 2 | 4, 3 | 4, 4 | 4, 5 | 4, 6 |
5 | 5, 1 | 5, 2 | 5, 3 | 5, 4 | 5, 5 | 5, 6 |
6 | 6, 1 | 6, 2 | 6, 3 | 6, 4 | 6, 5 | 6, 6 |
What is the probability that a roll includes a $2?$
What is the probability that a roll includes a 2 or results in a pair?
What is the probability that a roll doesn’t include a 2 or result in a pair?
$\frac{5}{9}$
What is the probability of rolling a 5 or a 6?
What is the probability that a roll includes neither a 5 nor a 6?
$\frac{4}{9}$
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)
What is the percent chance that all the children attending the party prefer soda?
What is the percent chance that at least one of the children attending the party prefers milk?
$1-\frac{C\left(350,8\right)}{C\left(500,8\right)}\approx 94.4\%$
What is the percent chance that exactly 3 of the children attending the party prefer soda?
What is the percent chance that exactly 3 of the children attending the party prefer milk?
$\frac{C\left(150,3\right)C\left(350,5\right)}{C\left(500,8\right)}\approx 25.6\%$
Write the first four terms of the sequence defined by the recursive formula $a=\u201314,{a}_{n}=\frac{2+{a}_{n\u20131}}{2}.$
$-14,-6,-2,0$
Write the first four terms of the sequence defined by the explicit formula ${a}_{n}=\frac{{n}^{2}\u2013n\u20131}{n!}.$
Is the sequence $0.3,\text{}1.2,\text{}2.1,\text{}3,\dots $ arithmetic? If so find the common difference.
The sequence is arithmetic. The common difference is $d=\mathrm{0.9.}$
An arithmetic sequence has the first term ${a}_{1}=-4$ and common difference $d=\u2013\frac{4}{3}.$ What is the 6 ^{th} term?
Write a recursive formula for the arithmetic sequence $-2,\text{}-\frac{7}{2},\text{}-5,\text{}-\frac{13}{2},\dots $ and then find the 22 ^{nd} term.
${a}_{1}=-2,\text{}{a}_{n}={a}_{n-1}-\frac{3}{2};\text{}{a}_{22}=-\frac{67}{2}$
Write an explicit formula for the arithmetic sequence $15.6,\text{}15,\text{}14.4,\text{}13.8,\dots $ and then find the 32 ^{nd} term.
Is the sequence $\text{\hspace{0.17em}}-2\text{,}-1\text{,}-\frac{1}{2}\text{,}-\frac{1}{4}\text{,}\dots $ geometric? If so find the common ratio. If not, explain why.
The sequence is geometric. The common ratio is $r=\frac{1}{2}.$
What is the 11 ^{th} term of the geometric sequence $\text{\hspace{0.17em}}-1.5,-3,-6,-12,\dots ?$
Write a recursive formula for the geometric sequence $1,\text{}-\frac{1}{2},\text{}\frac{1}{4},\text{}-\frac{1}{8},\dots $
${a}_{1}=1,\text{}{a}_{n}=-\frac{1}{2}\cdot {a}_{n}{}_{-1}$
Write an explicit formula for the geometric sequence $4,\text{}-\frac{4}{3},\text{}\frac{4}{9},\text{}-\frac{4}{27},\dots $
Use summation notation to write the sum of terms $\text{\hspace{0.17em}}3{k}^{2}-\frac{5}{6}k\text{\hspace{0.17em}}$ from $\text{\hspace{0.17em}}k=-3\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}k=15.$
$\sum _{k=-3}^{15}\left(3{k}^{2}-\frac{5}{6}k\right)$
A community baseball stadium has 10 seats in the first row, 13 seats in the second row, 16 seats in the third row, and so on. There are 56 rows in all. What is the seating capacity of the stadium?
Use the formula for the sum of the first $n$ terms of a geometric series to find $\sum _{k=1}^{7}-0.2\cdot {\left(-5\right)}^{k-1}.$
${S}_{7}=-2604.2$
Find the sum of the infinite geometric series $\sum _{k=1}^{\infty}\frac{1}{3}\cdot {\left(-\frac{1}{5}\right)}^{k-1}}.$
Rachael deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned?
Total in account: $\text{\$}140,355.75;$ Interest earned: $\text{\$}14,355.75$
In a competition of 50 professional ballroom dancers, 22 compete in the fox-trot competition, 18 compete in the tango competition, and 6 compete in both the fox-trot and tango competitions. How many dancers compete in the fox-trot or tango competitions?
A buyer of a new sedan can custom order the car by choosing from 5 different exterior colors, 3 different interior colors, 2 sound systems, 3 motor designs, and either manual or automatic transmission. How many choices does the buyer have?
$5\times 3\times 2\times 3\times 2=180$
To allocate annual bonuses, a manager must choose his top four employees and rank them first to fourth. In how many ways can he create the “Top-Four” list out of the 32 employees?
A rock group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if have 15 songs to pick from?
$C\left(15,3\right)=455$
A self-serve frozen yogurt shop has 8 candy toppings and 4 fruit toppings to choose from. How many ways are there to top a frozen yogurt?
How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E?
$\frac{10!}{2!3!2!}=151\text{,}200$
Use the Binomial Theorem to expand ${\left(\frac{3}{2}x-\frac{1}{2}y\right)}^{5}.$
Find the seventh term of ${\left({x}^{2}-\frac{1}{2}\right)}^{13}$ without fully expanding the binomial.
$\frac{429{x}^{14}}{16}$
For the following exercises, use the spinner in [link] .
Construct a probability model showing each possible outcome and its associated probability. (Use the first letter for colors.)
What is the probability of landing on an odd number?
$\frac{4}{7}$
What is the probability of landing on blue?
What is the probability of landing on blue or an odd number?
$\frac{5}{7}$
What is the probability of landing on anything other than blue or an odd number?
A bowl of candy holds 16 peppermint, 14 butterscotch, and 10 strawberry flavored candies. Suppose a person grabs a handful of 7 candies. What is the percent chance that exactly 3 are butterscotch? (Show calculations and round to the nearest tenth of a percent.)
$\frac{C\left(14,3\right)C\left(26,4\right)}{C\left(40,7\right)}\approx 29.2\%$
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