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How many subsets does the set { 1 ,   3 ,   5 ,   ,   99 } have?

2 50 = 1.13 × 10 15

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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?

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How many distinct ways can the word DEADWOOD be arranged?

8 ! 3 ! 2 ! = 3360

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How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?

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Binomial Theorem

Evaluate the binomial coefficient ( 23 8 ) .

490 , 314

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Use the Binomial Theorem to expand ( 3 x + 1 2 y ) 6 .

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Use the Binomial Theorem to write the first three terms of ( 2 a + b ) 17 .

131 , 072 a 17 + 1 , 114 , 112 a 16 b + 4 , 456 , 448 a 15 b 2

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Find the fourth term of ( 3 a 2 2 b ) 11 without fully expanding the binomial.

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Probability

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
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What is the probability that a roll includes a 2 ?

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What is the probability of rolling a pair?

1 6

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What is the probability that a roll includes a 2 or results in a pair?

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What is the probability that a roll doesn’t include a 2 or result in a pair?

5 9

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What is the probability of rolling a 5 or a 6?

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What is the probability that a roll includes neither a 5 nor a 6?

4 9

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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?

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What is the percent chance that at least one of the children attending the party prefers milk?

1 C ( 350 , 8 ) C ( 500 , 8 ) 94.4 %

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What is the percent chance that exactly 3 of the children attending the party prefer soda?

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What is the percent chance that exactly 3 of the children attending the party prefer milk?

C ( 150 , 3 ) C ( 350 , 5 ) C ( 500 , 8 ) 25.6 %

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Practice test

Write the first four terms of the sequence defined by the recursive formula a = 14 ,   a n = 2 + a n 1 2 .

14 , 6 , 2 , 0

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Write the first four terms of the sequence defined by the explicit formula a n = n 2 n 1 n ! .

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Is the sequence 0.3 ,   1.2 ,   2.1 ,   3 , arithmetic? If so find the common difference.

The sequence is arithmetic. The common difference is d = 0.9.

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An arithmetic sequence has the first term a 1 = 4 and common difference d = 4 3 . What is the 6 th term?

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Write a recursive formula for the arithmetic sequence 2 ,   7 2 ,   5 ,   13 2 , and then find the 22 nd term.

a 1 = 2 ,   a n = a n 1 3 2 ;   a 22 = 67 2

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Write an explicit formula for the arithmetic sequence 15.6 ,   15 ,   14.4 ,   13.8 , and then find the 32 nd term.

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Is the sequence 2 , 1 , 1 2 , 1 4 , geometric? If so find the common ratio. If not, explain why.

The sequence is geometric. The common ratio is r = 1 2 .

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What is the 11 th term of the geometric sequence 1.5 , 3 , 6 , 12 , ?

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Write a recursive formula for the geometric sequence   1 ,   1 2 ,   1 4 ,   1 8 ,

a 1 = 1 ,   a n = 1 2 a n 1

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Write an explicit formula for the geometric sequence 4 ,   4 3 ,   4 9 ,   4 27 ,

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Use summation notation to write the sum of terms 3 k 2 5 6 k from k = 3 to k = 15.

k = 3 15 ( 3 k 2 5 6 k )

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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?

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Use the formula for the sum of the first n terms of a geometric series to find k = 1 7 0.2 ( 5 ) k 1 .

S 7 = 2604.2

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Find the sum of the infinite geometric series k = 1 1 3 ( 1 5 ) k 1 .

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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: $ 140 , 355.75 ; Interest earned: $ 14 , 355.75

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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?

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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 × 3 × 2 × 3 × 2 = 180

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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?

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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 ( 15 , 3 ) = 455

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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?

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How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E?

10 ! 2 ! 3 ! 2 ! = 151 , 200

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Use the Binomial Theorem to expand ( 3 2 x 1 2 y ) 5 .

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Find the seventh term of ( x 2 1 2 ) 13 without fully expanding the binomial.

429 x 14 16

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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.)

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What is the probability of landing on an odd number?

4 7

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What is the probability of landing on blue?

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What is the probability of landing on blue or an odd number?

5 7

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What is the probability of landing on anything other than blue or an odd number?

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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.)

C ( 14 , 3 ) C ( 26 , 4 ) C ( 40 , 7 ) 29.2 %

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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