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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

How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
Practice Key Terms 9

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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