Permutations and Combinations Questions and Answers

About the sequence x keZ we have the following a x 1 B x b for each 1 to b t b MKk X 1 x 1 7 te to b t b k20 k 1 c x keZ is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 8B x b C which is given by 1 l b t b b Thus x 1 lim x ff s x s ds 5 8 8 10 de x lim ff s x s ds x flim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function rx 1 satisfying the integral equation x 1 x ff s x s ds for all te to b t b xt lim x 1 Finally the validity of this integral equation has the following two implications X t x s x s ds x 0 x 1 2 dff s x s ds f 1 X t This now proves that the curve X 1 8 1 82 thus obtained is a solution of the initial value problem dx dt 0

About the sequence x keZ we have the following a x 1 B x b for each t to b t b MKk b x 1 x 1 1 to b t bk20 k 1 c xk Z is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 to 8 B x b Cn which is given by x t lim x 1 t t b to b Thus x 1 lim xo f s x s ds 168 x 10 de 1 Finally the validity of this integral equation has the following two implications X t x f s x s ds x 0 xo x lim ff s x s ds x lim f s x s ds a kix x f f s lim x s ds x ff s x s ds Thus the function rx 1 satisfying the integral equation x 1 x f f s x s ds for all 1 t b 1 b 2 dx dt aff s x s ds f 1 X 1 This now proves that the curve X 1 8 4 82 thus obtained is a solution of the initial value problem

To say that 5 4 is the same as saying z is in the closed interval A B where A is and where B is

Select the correct choices that complete the sentence below The value of cos 390 is because 390 is in quadrant value of cos 390 is The reference angle is a

Suppose ABC is a right triangle with sides a b and c and right angle at C Use the Pythagorean theorem to find the unknown side length Then find the values of the six trigonometric functions for angle B Rationalize the denominators when applicable a 5 c 6 C The unknown side length b is 7 61 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression

Solution To get the sampling distribution of probability value we should first determine whether the Sample is sufficient using the following two conditions npz 10 n 1 P 10 mp 150x 0 25 37 5210 The two conditions are fulfilled We understand from the central limit theorem that the as being is roughly normally distributed with the mean up p and the standard deviation PFP along with of 7 npzlo and n 1 P 10 Therfore the 2 conditions which satisfy the proportion of sample that were normally distributed by the mean up P and the Standard deviation PCHPJJ o p Sample proportion mean is up P 0 25 The sample proportion of the sitem and deviation of op 3 0 25 0 25 150 0025 0 0625 P 1 P n J 10 1875 0 00125 150 0 035 pproximately normer MP 0 25 op 0 035 Hence is correct option

roblem 5 If a square matrix A satisfies 46 54 41 0 which of the following is equal to 4 A 51 4 B 51 46 C 41 4 D 51 4 E 51 A6 F 41 A G 51 A H 61 A Problem 5 Select v

About the sequence x keZ we have the following a x 1 B x b for each 1 to b to b MK b x 1 x1 1 to b t b k 20 k 1 c x keZ is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 to 8 B x b which is given by x t lim x 1 11 b t b Thus X 1 Jim A 506 8 5 de Finally the validity of this integral equation has the following two implications X 1 xo f s x s ds x 0 x 1 x lim ff s x s ds x flim f s x s ds x f f s lim x s ds xo ff s x s ds Thus the function tx 1 satisfying the integral equation x 1 x f s x s ds for all t 16 b 1 b 2 dt 0 ff s x s ds f t X t This now proves that the curve X 1 6 82 thus obtained is a solution of the initial value problem

MATHEMATICS Proposition 3 Let x J y J be two solutions of the initial value problem dx f t x x t x dt then x t y t for all t JOJ Proof Recall that both x and y being solutions of the initial value problem satisfy the integral equations on their domain intervals x 1 x f s x s ds y 1 x ff s y s ds Therefore x 1 y 1 s x s s y s ds which implies x 0 y 0 0 S 5 x s s y s 0 Kx s y s for all t2te Applying Gronwalls result with A 0 we get 0 x 1 y 1 0 for all t t This gives the desired equality x 1 y t for all tj Towards the uniqueness of the solution of the initial value problem 3 we consider all the solutions of the initial value problem 3 Let the totality of them be denoted by x J XEA the solutions x being thus indexed by a suitable indexing set A Above we have verified that any two solutions say x and xx are equal on the overlap of their domains Therefore we patch together all the solutions to get a maximal solution is the solution defined on the largest open interval It is obtained as follows Let J U J A clearly Jis an open sub internal of I with x EJ and all the solutions x patch up to get a solution x J 2 of the initial value problem 3

Find the equation of the line with the slope 5 and the y intercept 0 8 8 The equation of the line is Type an equation Type your answer in slope intercept form Use integers or fractions for any numbers in the equatic

Find the equation of the line whose slope is 0 and y intercept is 0 12 Graph the line The equation of the line is Type an equation Type your answer in slope intercept form

Find the equation of the line whose slope is 4 and y intercept is 0 1 Graph the line The equation of the line is Type your answer in slope intercept form

Graph the line 8x 2y 4 using the slope and y intercept Step by Step Solution Step 1 Solve the equation for y to write it in slope intercept form y mx b 8x 2y 4 2y 8x 4 y V 8x 4 2 Subtract 8x from both sides Divide both sides by 2 Simplify

For the following equation find the missing value in each ordered pair x 6 a 1 a 1 b 11 c 13

Find the equation of the line described below Write the equation in slope intercept form if possible Graph the line Contains 2 8 and 3 7 Choose the correct answer below and fill in the answer box within your choice Simplify your answer Use integers or fractions for any numbers in the equation O A The equation of the line cannot be written in slope intercept form The equation of the line is OB The equation of the line in slope intercept form is

AX O A No the graphs are not equivalent Their edges are connected in different ways OB No the graphs are not equivalent One has a triangular shape and the other has a rectangular shape OC Yes the graphs are equivalent They have the same number of vertices connected to each other in the same way D Yes the graphs are equivalent Although their edges are connected in different ways they have the same numbe Watch the video and then solve the problem given below Click here to watch the video Are the two graphs equivalent Choose the correct answer below

Find the equation of the line that contains the given points Write the equation in slope intercept form if possible 1 3 and 0 6 Select the correct choice below and if necessary fill in the answer box to complete your choice Simplify your answer Use integers or fractions for any numbers in the equation OA The equation of the line in slope intercept form is OB The equation of the line cannot be written in slope intercept form The equation of the line is

Total number of drivers 1700 To find 99 Confidence interval for the population meam of drivers that has received at least 1 ticket Solution It is given that 33 Hence P 0 33 The CI we know that CI P Zx 1 P Now Substitude values in above furmula GI 0 33 2 576 0 33 1 0 33 1700 CI survey is of 1700 drivers hence n 1700 drivers have received ticke 0 33 2 576 1 033x 0 67 1700 0 332 576 x 0 011 40 1 0 33 0 0294 Now we construct the confidence interval 0 33 0 0294 2 CI 0 33 0 0294 0 3006 CI 0 3594 CI 30 06 35 941 Answer Hence correct option is A

The theater society members are voting for the kind of play they will perform next semester a comedy C a drama D or a musical M Their votes are summarized in the following preference table Which type of play is selected using the plurality method A Number of Votes 15 12 First Choice Second Choice Third Choice is selected for next semester drama 12 M C D C M C D D M comedy 8 C D M 4 D M C 4 M D C CID

Use the equation to complete the table Use the table to list some of the ordered pairs that satisfy the equation y 4x 1 X 2 1 y x y X 2 1 4 y

Determine an ordered pair that satisfies the following equation by substituting the given value of the variable into the equation 2x 5y 9 y 1 The ordered pair is

Part 1 of 4 Find an ordered pair that satisfies the equation 2x y 11 30 O Step by Step Solution Step 1 Choose any value for one of the variables in the equation Choose any value of x or y For example let x 3

Consider the following sets U a b c d e f g h i j k A a d f h i k B b d f g h j Find the following 7 An B 8 ANB 9 An A 10 AU A

3 Use the distance formula to show that the points 1 5 0 3 and 2 1 are collinear on a line

2x 1 f x 3 x 21 FIND f 1 Type a response x 1 1 x 5 x 5

The parent function f x 2x is transformed into the function g x 4 2 x 3 8 Identify the transformations that occurred Vertical shift Up 8 Vertical shift Down 8 Horizontal shift Left 3 Horizontal shift Right 3 X Axis Reflection Vertical Stretch narrower Vertical Compression shrink y abx h k H Shift X reflect Left Right a 1 stretch o a 1 shrink compress N Shift Up Down Horiz Asymptote y

3 Look at the data and 1 identify the independent and dependent variables 2 determine if it can be expressed as a function and if it is a function 3 is it linear or nonlinear a b C Time t 0 Position s 3 Time t 5 Position s 7 Hour Calories 5 Force Pressure 2 1200 100 200 4 1 7 6 12 6 1000 500 10 807 2 4 7 800 750 15 34 8 6 8 600 800 16 4 6 9 2 8 20 400 1000 2500 be a function of F Can F be a function of P 50

In the following exercise a formula is given along with the values of all but one of the variables in the formula Find the value of the variable that is not given y mx c m 8 x 4 c 6 6

2 The annual salaries of all employees at a financial company are normally distributed with a mean of 34 000 and a standard deviation of 4 000 What is the z score of a company employee who makes an annual salary of 42 000 A 2 B 1 C 1 D 2

Solve y 6 8 y 6 7 0 Step by Step Solution Step 1 Determine the appropriate substitution and write the equation in the form au Let uay 6 y 6 8 y 6 7 0 8 1 7 0

Find the equation of the quadratic function f whose graph is shown below H 2 5 5 4 30 5 f x 0

Determine a formula for the quadratic function in vertex form that has its vertex at 7 4 and passes through the point 9 20 A y x 7 4 B y 4 x 9 20 C y 4 x 7 4 D y x 9 20 E y 4 x 7 4

The length of a rectangle is 10 yd less than three times the width and the area of the rectangle is 77 yd Find the dimensions of the rectangle Length yd Width d 010 X

Use exact numbers y x 9 8 7 6 5 4 3 2 9 876 6 6 SK 5 4 3 2 14 23 Y 2 8788 3 4 5 6 7 8 9 HE 1 2 3 4 5 6 7 8 9

5 If f x 2x 3 g x x x 21 find x when a f x 7 b g x

A jogging track has a length of 1496 yards yd How long is this in miles mi First fill in the blank on the left side of the equation using one of the ratios Then write your answer on the right side of the equation Ratios 1 ft 12 in 1496 yd 12 in 1 ft 1 mi 5280 ft 0 5280 ft 1 mi 1760 yd 1 mi 1 mi 1760 yd ft 00 X yd in S mi

olve the equation For an equation with real solutions support the answer graphically x 8x 9 ewrite the equation in factored form E x 9 x 1 0 Factor completely he solution set is Type an integer or a simplified fraction Type an exact answer using radicals and i as needed Use a c eparate answers as needed

Write the interval notation for this compound inequality Explain why parenthesis brackets etc Lesson 1 05 x 4 or x 8

Given the following function a find the vertex b determine whether there is a maximum or a minimum value and find the value c find the range and d find the intervals on which the function is increasing and the intervals on which the function is decreasing f x 4x 4x 3 a The vertex is Type an ordered pair using integers or fractions

Graph the linear function using the slope and y intercept y x 3 10

Use the table below to write the Mayan numeral as a Hindu Arabic numeral 5 10 15 6 11 16 O 19 O 10 7 O 14 12 17 3 8 13 18 g 14 19

If the Babylonian numeral v stands for one and the Babylonian numeral stands for ten then write the Babylonian numeral as a Hindu Arabic numeral VVV 05 O 50 O 23 32

5 Kathy plans to purchase a car that depreciates at a rate of 12 per year The initial value of the car is 21 000 Use an exponential function to find the value of the car after 3 years

Thus x t lim xo f s x s ds Finally the validity of this integral equation has the following two implications X 1 x f s x s ds x 0 xg 1 x lim ff s x s ds x S lim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function rx 1 satisfying the integral equation x 1 x f f s x s ds for all 1 t b 1 b 2 dx dt 0 ff s x s ds f 1 X 1 This now proves that the curve X 1 8 82 thus obtained is a solution of the initial value problem

Chus 360 1 4 1 1 4 1 4 2 2 30 2 3 2 9 91 233723 2 1 3 3ds 3 3r 3 3t 1 5 1 2 x 1 300 2 6 8s gi 91 441 11 1 32 33 8 4 91 sds 1 3 91 44 19 44 2 3 62 3 2 22 3

x 1 2 2 2 3 6 2 191 441 62 3 3 3t 3 8 91 44 19 44 3 62 3 3 1 2 9s s ds 91 2 91 1 17 2 5 2 4 I 3 6 2 1 3 x s z s ds 41 191 32 3 I 44 4s 4 3s ds 27 3 4 2 8 271 41 4 16 4 27

Use the appropriate reciprocal identity to find the exact value of cot 8 for the given value of tan e tan 0 1 25 cot 0 Type an intener or a dosin

Explain why the two figures show equivalent graphs Then draw a third equivalent graph D H M M Why are the two graphs equivalent Select all that apply A The graphs have the same number of vertices B The graphs have a number of edges that matches the number of vertices c The vertices are connected in the same way D The graphs have the same number of edges D M H N

8 2 1 4 5 5

Suppose that you decide to borrow 17 000 for a new car You can select one of the following loans each requiring regular monthly paymen Installment Loan A three year loan at 5 1 Installment Loan B five year loan at 4 8 PA Use PMT to complete parts a through c below The total interest for Loan B is 2155 6 Round to the nearest cent as needed c Compare the monthly payments and the total interest for the two loans Determine which loan is more economical Choose the correct answer below A The three year loan at 5 1 is more economical B The five year loan at 4 8 is more economical The buyer will save approximately in interest