Binomial theorem Questions and Answers

By expanding e tx Expand e in a Maclaurin series exx tx 2 in a Maclaurin series and integrating term by term show that My 1 f x dx 1 l I f H 2 e 8 tx 2x 2 te tx 1 te 2 2x 2 2x 21 2 2x te 2 t x r t e r t x r t ex r

2 By expanding e in a Maclaurin series and integrating term by term show that Mx 1 f x dx 1 ut 2 tx Expand e in a Maclaurin series tx tx 22 00 XX 81

A board game uses the shown to the right Two cards are selected 14 at random from this deck Determine the probability that neither card shows a 1 or a 4 both with and without replacement 12 N R 3 23 R 4 10 5 2 3 4 5 2 3 4 5 Determine the probability that neither card shows a 1 or a 4 with replacement Type an integer or a simplified fraction

Experiments done without replacement will result in

Number of students m 2 2 Annual Car Insurance Premiums for Students 650 701 752 803 854 905 956 1007 a How many students were surveyed There were students who were surveyed

A system proposed by the U S Navy for underwater submarine communication called ELF Extremely Low Frequency operates with a frequency of 76 Hz 1000 m 1 km 1km 0 62137 mi C AV E hv c 3 00 x 108 m s h 6 626 x 10 34 J s 1 Hz 1 s1 HINT Be sure to report final answers to correct significant figures units and in scientific notation c What is the energy of radiation in joules 5 0 x 10 32 5 03 x 10 32 8 72 x 10 36 8 7 x 10 36

Five people plan to meet after school and if they all show up there will be one group of five people However if only three of them show up in how many ways is this possible ways

Where will the median be located for this distribution Mode Mean mode median mean median mode median mean median mean

In the expression number of failures 3 4 7 10 23 0 6 0 4 7 10 which number represents the

blank I rrect answer from the choices listed below and use it to fill in the This expression is equivalent to 10C3 Probability that is based on a best guess given personal experience This type of calculation is used when order doesn t matter A possible result in a probability experiment A variable that can take on only certain values within a given range For example there can be no decimals The number of ways of selecting 4 dogs or 3 cats from a group of 10 dogs and 5 cats An expression equivalent to 5 x 4 The number of ways of selecting 4 dogs and 3 cats from a group of 10 dogs and 5 cats This is the probability of rolling a sum of 9 with a pair of die A distribution with independent trials whose outcomes are either success or failure 1 subjective probability 2 outcome 3 4 5 12 3 continuous 6 permutation 10 3 7 binomial 8 5 9 1 12 10 combination 11 1 9 12 experimental probability 13 hypergeometric 14 10 3 16 15 discrete 10 5 90 4 3

A pizza shop offers a special on a 2 topping pizza from a list of three items The chart below shows all possible pizza combinations Pizza Topping 1 Pepperoni Ham Sausage Pepperoni PP PH PS Ham HP HH HS Sausage SP SH SS If a person randomly chooses the items on his pizza what is the probability he chooses a pizza with double pepperoni Pizza Topping 2 Select one 11 12 O a b 1 C 1 3

inomial s tossed 484 times Use the normal curve approximation to find the probability of obtaining reen 226 and 254 heads inclusive ctly 245 heads er than 216 or more than 269 heads probability is to four decimal places as needed Standard Normal Distribution Table Page 2 08 09 07 00 05 04 03 0 5359 02 0 5319 0 5279 01 00 0 5160 0 5199 0 5239 0 5080 0 5120 0 5040 0 5636 0 5675 0 5714 0 5753 0 5596 0 0 0 5000 0 5517 0 5557 0 5478 0 6103 0 6141 0 6064 0 0020 0 5 438 0 5398 0 5087 0 1 0 5910 0 5948 0 6480 0 6517 0 6443 0 6406 0 5832 0 5871 0 6368 0 5793 0 6879 0 2 0 6331 0 6293 0 6844 0 6179 0 6217 0 6255 0 3 0 6700 0 6736 0 6772 0 6808 0 6664 0 0028 0 6501 0 4 0 6554 0 7019 0 7054 0 7088 0 7123 0 7157 0 7190 0 7224 0 6985 0 6950 0 6915 0 5 0 7324 0 7357 0 7389 0 7422 0 7454 0 7486 0 7517 0 7549 0 7291 0 8133 0 6 0 7257 0 8078 0 8106 0 7642 0 7673 0 7704 0 7731 0 7764 0 7794 0 7823 0 7852 0 7611 0 7 0 7580 0 8340 0 8365 0 8389 0 7939 0 7967 0 7995 0 8023 0 8051 0 7910 0 8 0 7881 0 8621 0 8599 0 8577 0 8212 0 8238 0 8261 0 8289 0 8315 0 8186 0 8554 0 8159 0 8830 0 8810 0 8770 0 8790 0 9015 0 8997 0 8413 0 8062 0 8980 0 9162 0 9 1 0 1 1 0 8438 0 8065 0 8485 0 8508 0 8531 0 8729 0 8749 0 8708 0 8643 1 2 0 8849 0 8869 0 8461 0 8686 0 8925 0 8944 0 8888 0 8007 0 9115 0 9082 0 9099 0 9265 0 9251 0 9236 1 3 0 9032 0 0049 0 9131 0 9147 0 9177 0 500 0 9222 0 9192 0 9207 0 9279 0 9292 0 9306 0 9319 1 4 0 9345 0 9332 1 5 1 0 0 9452 0 9463 0 9357 0 9370 0 9382 0 9394 0 9406 09118 0 9429 0 9441 0 9474 0 9484 0 9495 0 9505 0 9515 0 9525 0 9616 0 9573 0 9582 0 9501 0 9608 1 7 1 8 1 9 0 9656 0 9064 0 9671 0 9554 0 9564 0 9649 0 9641 0 9719 0 9713 2 0 0 9772 0 9778 0 9732 0 9726 0 9738 0 9545 0 9535 0 9633 0 9625 0 9699 0 9093 0 9686 0 9761 0 9756 0 9750 0 9812 0 9803 0 0808 0 9798 0 9793 0 9854 0 9850 0 9783 0 9842 0 9846 0 9890 0 9887 0 9884 0 9881 0 9830 0 9838 0 9910 0 9913 0 9875 0 9878 0 9868 0 9909 0 9911 0 9906 0 9936 0 9934 0 9904 0 9032 0 9031 0 9029 0 9227 0 9788 0 9817 0 9857 0 0834 0 9871 0 9898 0 9901 0 9922 0 9925 0 9945 0 9946 0 9959 0 9900 2 1 2 2 2 3 2 4 2 5 2 6 0 9826 0 9821 0 9864 0 9861 0 9896 0 9893 0 9918 0 9020 0 9938 0 9940 0 9953 0 9955 0 9941 0 9943 0 9957 0 9956 0 9599 0 9678 0 9744 0 9948 0 9061 0 9949 0 9062 0 9951 0 9963 2 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 0 9706 0 9767 X 0 9952 0 9064 1 5 1 6 1 7 1 85 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 dard Normal Distribution Table Page 08 0 07 1 06 05 04 03 02 0 00 01 00 0 0003 0 0003 0 0003 0 0003 0 0003 0 0003 0 0004 0 00 0 0003 0 0003 0 0003 0 0004 0 0004 34 0 0004 0 0004 0 0004 0 00 0 0005 0 0005 0 0005 0 0005 0 0005 0 0006 3 3 0 04 0 0007 0 0006 0 0006 0 0006 0 0006 0 0008 0 0007 0 0007 3 2 0 0009 0 0008 0 0008 0 0008 0 0 0 0011 0 0010 3 1 0 0010 0 0009 0 0009 0 0 0 0014 0 0013 0 0013 0 0012 0 0012 0 0011 0 0011 0 0013 3 0 0 0016 0 0016 0 0015 0 0015 0 0 0 0021 0 0020 0 0019 0 0018 0 0018 0 0017 2 9 0 0023 0 0022 0 0021 0 0 0 0026 0 0025 0 0024 0 0023 2 8 0 0030 0 0029 0 0028 0 0027 0 0 2 7 0 0035 0 0034 0 0033 0 0032 0 0031 0 0040 0 0039 0 0038 0 0037 2 6 0 0054 0 0052 0 0051 0 0049 0 1 0 0047 0 0045 0 0044 0 0043 0 0041 0 0057 0 0055 0 00GG 0 25 2 5 0 0002 0 0060 0 0059 0 0069 0 0068 0 0 0073 0 0071 0 0087 0 0089 0 0091 2 4 0 0082 0 008s0 0 0078 0 0075 0 0 0096 0 0094 0 0099 0 0102 0 0116 0 0113 0 2 3 0 0107 0 0104 0 0125 0 0122 0 0119 0 0129 2 2 0 0139 0 0136 0 0132 0 0 0174 0 0170 0 0166 0 0162 0 0158 0 0151 0 0150 0 0146 0 0179 2 1 C 0 0222 0 0217 0 0212 0 0207 0 0202 0 0197 0 0192 0 0158 2 0 0 0228 0 0301 C 0 0281 0 0274 0 0268 0 0262 0 0256 0 0250 0 0244 0 0239 0 0314 0 0307 0 0329 0 0322 1 9 0 0287 0 0350 0 0351 0 0344 0 0336 1 8 0 0418 0 0409 0 0401 0 0392 0 0384 0 0375 0 0475 0 0465 0 0436 0 0427 0 0446 1 7 0 0505 0 0495 0 0485 0 0582 0 0571 0 0537 0 0526 0 0516 0 05 48 1 6 0 0630 0 0618 0 0006 0 0504 0 0655 0 06 13 0 0708 0 06941 0 0721 1 5 0 00GS 0 0764 0 0749 0 0735 0 0853 0 0838 0 0778 0 0800 0 0001 0 0885 1 4 0 0808 0 0793 0 0951 0 0034 0 0918 1 3 0 0968 0 1251 0 1230 0 1210 0 1190 0 1151 0 1131 0 1112 0 1093 0 1075 0 1056 0 1038 0 1020 0 1003 1 2 0 1335 0 1314 0 1292 0 1271 0 1357 11 0 1587 0 1562 0 1539 0 1515 0 1492 0 1469 0 1446 0 1423 0 1401 1 0 Print Done Check

Susan is a sales representative who has a history of making a successful sale from about 74 of her sales contacts If she makes 12 successful sales this week Susan will get a bonus Let n be a andom variable representing the number of contacts needed for Susan to get the 12th sale a Explain why a negative binomial distribution is appropriate for the random variable n We have binomial trials for which the probability of success is p 0 74 and failure is q 0 26 k is a fixed whole number 2 1 We have binomial trials for which the probability of success is p 0 26 and failure is q 0 74 k is a fixed whole number 2 1 We have poisson trials for which the probability of success is p 0 74 and failure is q 0 26 k is a fixed whole number 1 O We have binomial trials for which the probability of success is p 0 74 and failure is q 0 26 k is a fixed whole number 1 Write out the formula for P n in the context of this application Use C a b as the notation for a choose b P n 1 X b Compute P n 12 P n 13 and P n 14 Use 4 decimal places P 12 0 0270 P 13 0 0842 P 14 0 1424 X x c What is the probability that Susan will need from 12 to 14 contacts to get the bonus Use 4 decimal places 0 2536 x d What is the probability that Susan will need more than 14 contacts to get the bonus Use 4 decimal places 0 7464 X e What are the expected value and standard deviation a of the random variable n Use 2 decimal places 3 87 x 2 39 Interpret these values in the context of this application O The expected contact in which the thirteenth sale occurs is with a standard deviation of a O The expected contact in which the twelfth sale occurs is o with a standard deviation of O The expected contact in which the eleventh sale occurs is with a standard deviation of a The expected contact in which the twelfth sale occurs is with a standard deviation of a

Find the first 4 terms in the binomial expansion of 1 3x Select one a 1 21x 189x 945x b 1 21x 189x2 945x c 1 21x 189x 2 945x 3 Od 1 21x 189x 945x

In a particular school vote 30 of all voters are girls If two voters are randomly selected for a survey find the probability that they are both girls Select one a 0 03 b 0 30 C 0 6 d 0 09

Directions Select the most correct answer for each of the questions below 14 1 point 4 is a positive b negative undefined none of the above

Suppose a runner claims that her mile run time is less than 9 minutes on average Several of her friends do not believe her so the runner decides to do a hypothesis test at a 10 significance level to persuade them She runs 12 time trials collects the proper data and works through the testing procedure Ha u 9 Ha R 9 a 0 1 significance level The test statistic is B 20 Select the correct answer below I po n 7 7 9 12 The critical value is 20 1 1 28 Conclude whether to reject or not reject Ho and interpret the results 4 5 Reject Ho At the 10 significance level the test results are not statistically significant and at best provide weak evidence against the null hypothesis Reject Ho At the 10 significance level the data provide sufficient evidence to conclude that the mean time is less than 9 minutes Do not reject Ho At the 10 significance level the test results are not statistically significant and at best provide weak evidence against the null hypothesis Do not reject Ho At the 10 significance level the data provide sufficient evidence to conclude that the mean time is less than 9 minutes

Find the sum of the first 14 odd positive integers

terms of T 1 G J and n the angle through which end A rotates

with replacement A fair die is rolled 6 times What is the probability of having no 1 and no 3 among the rolls Round your answer to three decimal places 0 X 00

2 3 pizzas 4 5 7 8 9 10 11 Izzo s Pizza sells four types of pizza crust Last week the owner tracked the number sold of each type and this is what he found Type of Crust Number Sold Thin crust 315 Thick crust 248 Stuffed crust 150 Pan style 181 12 13 Based on this information of the next 2500 pizzas he sells how many should he expect to be thick crust Round your answer to the nearest whole number Do not round any intermediate calculations

Find the perimeter of the triangle whose vertices are 6 3 6 1 and 9 6 Write the exact answer Do not round

6 1 point A school dance committee is to consist of 2 freshmen 3 sophomores 4 juniors and 5 seniors If 5 freshmen 7 sophomores 9 juniors and 8 seniors are eligible to be on the committee in how many ways can the committee be chosen Your answer is nauora submittadi

QUE How many permutations of letters HIJKLMNOP contain the strings PON and KH Give your answer in numeric form 1

Find the derivative of the function y x 1 5 x 2

Determine the sum of the Decagon The sum of the measures of the interior angles of a decagon is

How many possible outcomes would there be if you tossed two coins with each coin has one head and one tail and a numbered cube numbers 1 2 3 4 5 6 Hint the answer is a whole number less than 30 but greater than 20 O 18 outcomes O 36 outcomes O 12 outcomes O 24 outcomes

6 8 O It is yellow O 0 250 O 0 125 2 3 Figure 2 Numbered spinner with multiple colors ETA 4013 Using Figure 2 above what is the probability of the spinner stopping and the number 1 Answer to be in decimal format to the nearest 1000ths

3 Suppose starting from point A you must walk to the river shown in the picture then cross the river on a boat to drop off a package on the other side of the river anywhere on the other side will work then cross the river back and walk to point B to deliver another package there Determine the point C on the other side of the river where you need to drop off your package in order to minimize the total distance that you have to travel A 0 4 C 2 0 River

Permutation vs combination Which notation permutation r permutation or r combination would you use in each of the given problems V How many permutations of the letters ABCDEFGH contain the string A 5 combination ABC How many ways are there to select a first prize winner a second prize winner and a third prizewinner from 100 different people who have entered a contest How many ways are there to select five players from a 10 member tennis team to make a trip toa match at another school A group of 30 people has been trained as astronauts to go on the first mission to Mars How many ways are there to select a crew of six people to go on this mission assuming that all crew members have the same job B Permutation C 8 permutation D 3 permutation E 5 permutation F 6 combination G 3 combination

What is the coefficient of x 12 13 in the expansion 2x 3y 25 Select ALL that applies 0 M 25 13 O 25 13 212313 25 12 13 25 12 13 25 13 212313 212313

Choose all correct formulas to calculate r permutations and r combinations of n elements C n r n P n r r D P n r C n r n n r C n r r P n r n n 1 n 2 n r 1 n r n r

Estimate the answer by picking the best choice without calculating Which of the following is a reasonable APR for a credit card 9 1 2 40

Use the binomial theorem to expand 3x y2 4 Show your work on paper and then upload an image of your work using the button below Attach File

How many bit strings of length 7 have at most three 1s O99 35 64 29

Which of the following is a coefficient of x7 in 2 x 330 5280 5280 330

Some questions make use of the Interactive Graphing Tool which allows you to draw a variety of graphs Click the icon to view instructions for the Interactive Graphing Tool Draw a dashed line using the points 0 0 and 4 4 which are shown on the graph and shade the region of the graph above the dashed line Use the line tool button and the dashed selector button Click to enlarge graph on the graphing palette to 3 draw a dashed line using the given points on the graph Then use the region shading tool to place a shading bucket above the dashed line H 10 B 10 8 16 4 6 48 P 10 K

Last Saturday at Pasquale s Pizzas and Wings 65 customers were served over the course of the evening Fifty two customers ordered pizza and 16 ordered buffalo wings Twelve of these customers ordered both pizza and wings Let P those who ordered pizza and W those who ordered wings Use this information along with the information in the two way table below to answer the following questions Wings No Wings a Find P Wings Pizza 12 40 b Find P Wings and Pizza c Find P Wings Pizza d Find P Pizza Wings No Pizza 4 9 C

The converse of p q is The converse of p q is

1 Answer the following questions based on the histogram below Hours Spent Playing Sports Weekly 10 8 4 02 2 1 29 4 5 2 Create a histogram using the data set below of Books Read this Summer 8 3 5 28 7722 6 8 7 g 83 3 a How many students spend over 5 hours playing sports weekly 9 1 b What is the most common time students play sports each week of Books 0 3 4 6 7 9

An entire group of people you are conducting your experiment on is called O Sample Individual Census O Population

Anastasia categorized her spending for this month into four categories Rent Food Fun and Other The percents she spent in each category are pictured here Food 20 Other 36 Rent 28 Fun 16 a If Anastasia spent a total of 2500 this month how much did she spend on Food

1 2 3 PL K LLL Calculate the terms through order five of the power series expansion about z 0 of the function z sin z Solution 2 2 23 3 25 5 sin z 1 22 3 24 5 1 0 2 3 5 7 1 2 6 a4z a626 0 27 a4 6 Z sin z 77 3 36 3 5 31 1 2 724 360 120 10 3 360 0 26 7 360 6 1 14 4 5 7 140 78 3 7 3 7 62 317 312

A rectangular field measures 100m by 80m A path is to be built on the inside edge of the field with constant width of w If the path should take up 5 of the total area of the field how wide should it b 80m 100m WO

19 You need 2 letters and 3 numbers for your password If you are able to reuse the letters and numbers how many options are there to choose from O 6 O 676 000 260 98 800

1 2 3 4 5 FRACTION OF INCHES ONLY 33 7 16 33 437 24 7 32 44 5 8 11 INCHES ONLY IN DECIMAL FORM 24 475 94 625 816 15 6411 81 24 32 3 8 11 32 ste FEET INCHES AND FRACTIONS OF INCH 2 97 6 J 17 32 7 10 5 8 6 4 FEET ONLY IN DECIMAL FORM 2 786 204 7 885 6 7698

Use the binomial theorem or Pascal s triangle to expand 3x 2y You can either use the superscript for an exponent or can use to represent an exponent So and you BIU 11 POSSIBLE POINT

Key Features of Quadratic Relations LEARNING GOAL We are learning to identify and describe key properties of quadratic relations x a Examine the tables of values below and graph the data What type of relationship would best model each set 2 1 0 2 V 5 1 A quadratic function has an equation of the form y ax bx a 0 1 3 The graph shown here is the quadratic parent function of g x x which has a vertex at 0 0 5 10 8 10 8 6 4 2 6 4 2 2 4 a 6 8 10 2 4 6 8 10

Which of the following sets of data has the smallest mean Select one a 29 72 51 33 b 74 48 34 61 c 33 53 43 63 d 49 28 30 64

A polynomial c x with all real coefficients has two of its roots as 9 and 7 4i If n represents the degree of the polynomial what is the minimum value of n Answer