show that {(1,1,0),(1,0,1),(0,1,1)} is linearly independent subset of r^3

Answer:

Rita paid 47,500 dollars for the purchase price.

Step-by-step explanation:

We are given the sales tax on a new car is directly proportional to the purchase price of the car which means there is is something k such that

when you multiply it to the sales tax you get the purchase price.

Let's set this equation:

y=kx

Let y represent the purchase price and x the sales tax.

The second sentence tells us that (x,y)=(1500,30000).

We can plug this into y=kx to find the constant k.  (Constant means it stays the same no matter what the input and output is).

So we have:

30000=k(1500)

300    =k(15)      I went ahead and divided previous equation by 100.

Now divide both sides by 15:

300/15=k

Simplify:

20=k

So the equation to use the answer the question is

y=20x

where y is purchase price and x is sales tax.

So we want to know the purchase price on a car if the sales tax is 2375.

So replace x with 2375:

y=20(2375)

y=47500

Answer:

$47500

Step-by-step explanation:

If the amount of sales tax on a new car is directly proportional to the purchase price of the car and Victor bought a new car for $30,000 and paid $1,500 in sales tax and Rita bought a new car from the same dealer and paid $2,375 sales tax, Rita payed $47,500 for her car.

y=20(2375)

y=47500

Answer:  [tex]\dfrac{1}{100,000}[/tex]

Step-by-step explanation:

Given : The total number of digits in number system (0 to 9) = 10

The number of digits in a social security number = 9

After last four digits are printed, the number of digits remaining to print = 9-4=5

Since , repetition of digits is allowed, then the total number of ways to print 5 remaining digits is given by :-

[tex]10\times10\times10\times10\times10=100,000[/tex]

Now, the probability of getting the correct Social Security number of the person who was given the​ receipt is given by:-

[tex]\dfrac{\text{Number of correct code}}{\text{Total number of codes}}\\\\=\dfrac{1}{100,000}[/tex]

Final answer:

The probability of correctly guessing an entire Social Security number with the last four digits known is 0.001%, calculated by multiplying the probability of guessing each of the five unknown digits correctly, which is 1/10, resulting in (1/10)^5 or 1/100,000.

Explanation:

The question asks about the probability of correctly guessing an entire Social Security number given the last four digits. A Social Security number has nine digits and the digits can be repeated. If you know the last four digits, you need to guess the first five correctly.

Since each digit can be any number from 0 to 9, there are 10 possibilities for each digit. The probability of guessing one digit correctly is 1 out of 10 (1/10). To find the probability of guessing all five correctly, you need to multiply the probability for each digit, so the probability for all five is (1/10) x (1/10) x (1/10) x (1/10) x (1/10), which equals 1/100,000 or 0.00001. Therefore, the probability of getting the correct Social Security number is 0.00001 or 0.001%.

Answer:

Step-by-step explanation:

Let's rewrite this with dy/dx in place of y', since they mean the same thing.  But to solve a differential we will need to take the antiderivative by separation to find the general solution.

[tex]\frac{dy}{dx}+xy=x[/tex]

The goal is to get the x stuff on one side and the y stuff on the other side by separation.  But we have an xy term there that we need to be able to break apart.  So let's get everything on one side separate from the dy/dx and take it from there.

[tex]\frac{dy}{dx}=x-xy[/tex]

Now we can factor out the x:

[tex]\frac{dy}{dx}=x(1-y)[/tex]

And now we can separate:

[tex]\frac{dy}{(1-y)}=x dx[/tex]

Now we solve by taking the antiderivative of both sides:

[tex]\int\ {\frac{1}{1-y}dy }=\int\ {x} \, dx[/tex]

On the left side, the antiderivative of the derivative of y cancels out, and the other part takes on the form of the natural log, while we follow the power rule backwards on the right to integrate x:

[tex]ln(1-y)=\frac{1}{2}x^2+C[/tex]

That's the general solution.  Not sure what your book has you solving for.  Some books solve for the constant, C. Some solve for y when applicable.  I'm leaving it like it is.

Answer:

The probability that a randomly selected person is not color blind is 0.958

Step-by-step explanation:

Given,

C represents red-green color blindness,

Also, the probability that a randomly selected person is color blind,

P(C) = 0.042,

Thus, probability that a randomly selected person is not color blind,

P(C') = 1 - P(C) = 1 - 0.042 = 0.958

Answer:

a= 8 ways

b. 20

Step-by-step explanation:

Palindrome of length 6 means first three digits must be same as the last three in reverse. For example 123321 is palindrome of six digits.

There 2 bits 0 and 1

a.So each of first three digits can be filled in 2 ways

therefore, 2*2*2= 8 ways

number of different palindromes of 6 digits will be 8

b. In a 6 digit a palindrome there Are 6 spaces in which 3 spaces are to be filled with 1's

this cab be done in

[tex]_{6}^{3}\textrm{C}= \frac{6!}{3!\times3!}[/tex]

= 20

Answer: 0.8023

Step-by-step explanation:

Given : [tex]\text{Mean}=\mu=481 \text{ millimeters}[/tex]

[tex]\text{Standard deviation}=41 \text{ millimeters}[/tex]

Assuming these lengths are normally distributed.

The formula to calculate the z-score is given by :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x= [tex]516 \text{ millimeters}[/tex]

[tex]z=\dfrac{516-481}{41}=0.853658536585\approx0.85[/tex]

The p-value = [tex]P(z\leq0.85)=0.8023374\approx0.8023[/tex]

Hence, the required probability : 0.8023

Answer:

(2, 1)

Step-by-step explanation:

The best way to do this to avoid tedious fractions is to use the addition method (sometimes called the elimination method).  We will work to eliminate one of the variables.  Since the y values are smaller, let's work to get rid of those.  That means we have to have a positive and a negative of the same number so they cancel each other out.  We have a 2y and a 3y.  The LCM of those numbers is 6, so we will multiply the first equation by a 3 and the second one by a 2.  BUT they have to cancel out, so one of those multipliers will have to be negative.  I made the 2 negative.  Multiplying in the 3 and the -2:

3(-9x + 2y = -16)--> -27x + 6y = -48

-2(19x + 3y = 41)--> -38x - 6y = -82

Now you can see that the 6y and the -6y cancel each other out, leaving us to do the addition of what's left:

-65x = -130 so

x = 2

Now we will go back to either one of the original equations and sub in a 2 for x to solve for y:

19(2) + 3y = 41 so

38 + 3y = 41 and

3y = 3.  Therefore,

y = 1

The solution set then is (2, 1)

x=2 and y=1

proof:

-9x+2y=-16

-9(2)+2(1)=-16

which is a true statement

Answer:

D. Stanley Kubrick

Step-by-step explanation:

Answer:

[tex]\Large\textnormal{(D.) Stanley Kubrick}[/tex]

Step-by-step explanation:

Stanley Kubrick directed to Dr. Strangelove. I hope this helps, and have a wonderful day!

Answer:

The cumulative percentage of fathers who completed only elementary school is nearle 33%.

Step-by-step explanation:

Among 30 fathers:

You can fill these numbers into the table:

[tex]\begin{array}{cccc}&\text{Frequency}&\text{Cumulative frequency}&\text{Cumulative percentage}\\\text{Elementary school}&10&10&\approx 33\%\\\text{El. and high school}&10&20&\approx 67\%\\\text{El., high schools and college}&7&27&90\%\\\text{El., high, college and grad. sch.}&3&30&100\%\end{array}[/tex]

The cumulative percentage of fathers who completed only elementary school is nearle 33%.

Answer: 33%

Step-by-step explanation:

Answer: There are 4 people who only go to the game on Saturday.

Step-by-step explanation:

Let the number of people go on Saturday be n(A).

Let the number of people go on Sunday be n(B).

Since we have given that

n(A) = 8

n(B) = 12

n(A∪B)  = 16

According to rules, we get that

[tex]n(A)+n(B)-n(A\cap B)=n(A\cup B)\\\\8+12-n(A\cap B)=16\\\\20-n(A\cap B)=16\\\\n(A\cap B)=20-16=4[/tex]

So, n(only go on Saturday) = n(only A) = n(A) - n(A∩B) = 8-4 = 4

Hence, there are 4 people who only go to the game on Saturday.

Step-by-step explanation:

In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).  The remaining 40 cards are non-face cards.

The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.

a) Kyd's expected value is:

K = (12/52)(5) + (40/52)(-2)

K = -5/13

K ≈ -$0.38

b) North's expected value is:

N = (12/52)(-5) + (40/52)(2)

N = 5/13

N ≈ $0.38

c) Kyd is expected to lose money, and North is expected to gain money.  North has the advantage.

Kyd's expected value for this game is -$0.38.

North's expected value for this game is  $0.38.Kyd is expected to lose money, and North is expected to gain money.  

North has the advantage.

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

Given

In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).  The remaining 40 cards are non-face cards.

The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.

a) Kyd's expected value is:

K = (12/52)(5) + (40/52)(-2)

K = -5/13

K ≈ -$0.38

b) North's expected value is:

N = (12/52)(-5) + (40/52)(2)

N = 5/13

N ≈ $0.38

c) Kyd is expected to lose money, and North is expected to gain money.  North has the advantage.

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Answer:

C. 62 degrees

Step-by-step explanation:

Alright I see a circle and a half a rotation where the diameter is at. A half of 360 degrees is 180 degrees.

So the arc measure in degrees for EG is 180 degrees (both the left piece and right piece have this measure).

Since EG is 180 then FG=EG-EF=180-56=124.

To find x we have to half 124 since it is the arc measure where x is but x is the inscribed angle.

x=124/2=62

C.

62 degrees ! All the rest wouldn’t make sense (:

Answer: [tex]\dfrac{2}{1001}[/tex]

Step-by-step explanation:

Given : The number of female applicants = 9

The number of male applicants = 6

Total applicants = 15

The number of ways to select 5 applicants from 15 applicants :-

[tex]^{15}C_5=\dfrac{15!}{5!(15-5)!}=3003[/tex]

The number to select 5 applicants from 15 applicants such that no female applicant is selected:-

[tex]^{9}C_0\times^6C_5=1\times\dfrac{6!}{5!(6-5)!}=6[/tex]

Now, the required probability :-

[tex]\dfrac{6}{3003}=\dfrac{2}{1001}[/tex]

Answer:

To find out how much trim you would need for the window, you would need to calculate the Circumference.

Step-by-step explanation:

Circumference is the length around a circle and is calculated using the following formula.

[tex]C = 2\pi r[/tex]

In which C is the circumference and r is the radius. Since we already know the radius all we have to do is plug it into the equation and solve for C.

[tex]C = 2\pi (4.2)[/tex]

[tex]C = 26.389 ft[/tex]

Answer: The window has a Circumference of 26.389 feet so you would need that amount of trim to fit the area around the window.

Hope my answer would be a great help for you.    

If you have more questions feel free to ask here at Brainly.  

Approximately 27 feet of trim would be needed to go around a window with a radius of 4.2 feet.

To calculate the amount of trim needed to go around a round window with a radius of 4.2 feet, we can use the formula for the circumference of a circle. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. In this case, the radius is 4.2 feet. Plugging in the value of the radius into the formula, we get C = 2π(4.2) = 8.4π feet. Since we want the amount of trim needed, we can simply round the answer to the nearest whole number. Therefore, approximately 27 feet of trim would be needed to go around the window.

Answer:

The given statement is FALSE.

Step-by-step explanation:

If p1, p2, . . . , pn are prime, then A = p1p2 . . . pn1 + 1 is also prime.

No, this statement is false.

Let's take an example:

We take two prime numbers.

p1 = 3

p2 = 5

Now p1p2+1 becomes :

[tex](3\times5)+1=16[/tex]

And we know that 16 is not a prime number.

Note : A prime number is a number that is only divisible by 1 and itself like 3,5,7,11 etc.

Answer:5.8

Step-by-step explanation:

Given

weight of chair with person on it =2118 N

Barber applies a force of 63 N

Using pascal's law

we know pressure around both sides is equal

i.e. [tex]P_{chair}=P_{piston}[/tex]

pressure=[tex]\frac{force}{area}[/tex]

[tex]P_{chair}[/tex]=[tex]\frac{2118}{\pi \dot r_{plunger}^2}[/tex]

[tex]P_{piston}[/tex]=[tex]\frac{63}{\pi \dot r_{piston}^2}[/tex]

substituting values

[tex]\left [\frac{r_{plunger}^2}{r_{piston}^2}\right ][/tex]=[tex]\frac{2118}{63}[/tex]

[tex]\left [\frac{r_{plunger}}{r_{piston}}\right ][/tex]=[tex]\left [ \frac{2118}{63}\right ]^{0.5}[/tex]

[tex]\left [\frac{r_{plunger}}{r_{piston}}\right ][/tex]=5.798

Answer:

A (9, 3)

Step-by-step explanation:

First the point is rotated 90° counterclockwise about the origin.  To do that transformation: (x, y) → (-y, x).

So S(-3, -5) becomes S'(5, -3).

Next, the point is translated +4 units in the x direction and +6 units in the y direction.

So S'(5, -3) becomes S"(9, 3).

Answer:

The answer is [tex]\frac{11}{26}[/tex]

Step-by-step explanation:

Total number of cards in the deck = 52

Number of clubs = 13

Number of picture cards = 12

Number of picture cards that are clubs = 3

So, number of picture cards or clubs = [tex]13+12-3=22[/tex]

P(club or picture) = [tex]\frac{22}{52} =\frac{11}{26}[/tex]

The answer is [tex]\frac{11}{26}[/tex].

To determine the probability of selecting a club or a picture card from a standard deck of 52 cards, calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

To determine the probability that a card selected from a standard deck of 52 cards is a club or a picture card, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

There are 13 clubs and 12 picture cards (Jacks, Queens, and Kings) in a deck. However, we need to subtract the Queen of Clubs, as it has already been selected. So, the number of favorable outcomes is 13 + 12 - 1 = 24.

The total number of possible outcomes is 52, as there are 52 cards in a standard deck.

Therefore, the probability of selecting a club or a picture card is 24/52, which simplifies to 6/13 or approximately 0.4615.

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Answer:

The quotient is: x-7

The remainder is: -2

Step-by-step explanation:

We need to divide (x^2 - 13x +40) divided by (x- 6)

The quotient is: x-7

The remainder is: -2

The division is shown in the figure attached

Answer:

D. x = 9√3 and y = 18

Step-by-step explanation:

This is an isosceles triangle divided into two equal parts.

Step 1: 18 can be divided into 2 parts which makes the base of both triangles 9.

Step 2: Find the value of x

The value of x can be found through the tan rule.

tan (angle) = opposite/adjacent

tan (60) = x/9

√3 = x/9

x = √3 x 9

x = 9√3

Step 3: Find the value of y

The value of y can be found through the cos rule.

Cos (angle) = adjacent/hypotenuse

Cos (60) = 9/y

1/2 = 9/y

y = 18

Therefore, the correct answer is D; x = 9√3 and y = 18

!!

Answer: 24

Step-by-step explanation:

The formula to calculate the sample size is given by :-

[tex]n=(\frac{z_{\alpha/2}\ \sigma}{E})^2[/tex]

Given : Margin of error : [tex]E=0.5[/tex]

The significance level : [tex]\alpha=1-0.90=0.1[/tex]

Critical value : [tex]z_{0.05}=\pm1.645[/tex]

Standard deviation : [tex]\sigma=1.5[/tex]

Now, the required sample size will be :-

[tex]n=(\frac{1.645\times1.5}{0.5})^2=24.354225\approx24[/tex]

Hence, the required sample size = 24

Answer: 0.0018

Step-by-step explanation:

Binomial distribution formula :-

[tex]P(x)=^nC_xp^x(q)^{n-x}[/tex], here P(x) is the probability of getting success at x trial , n is the total number of trails, p is the probability of getting success in each trail.

Given : The probability that a child in the U.S. was living with both parents : p=0.70 ; q=1-0.70=0.30

If 25 children were selected at random in the U.S.,then the probability that at most 10 of them will be living with both of their parents will be :-

[tex]P(x\leq10)=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)+P(7)+P(8)+P(9)+P(10)\\\\=^{25}C_{0}(0.7)^0(0.3)^{25}+^{25}C_{1}(0.7)^1(0.3)^{24}+^{25}C_{2}(0.7)^2(0.3)^{23}+^{25}C_{3}(0.7)^3(0.3)^{22}+^{25}C_{4}(0.7)^4(0.3)^{21}+^{25}C_{5}(0.7)^5(0.3)^{20}+^{25}C_{6}(0.7)^6(0.3)^{19}+^{25}C_{7}(0.7)^7(0.3)^{`18}+^{25}C_{8}(0.7)^8(0.3)^{17}+^{25}C_{9}(0.7)^9(0.3)^{16}+^{25}C_{10}(0.7)^{10}(0.3)^{15}\\\\=(0.7)^0(0.3)^{25}+25(0.7)^1(0.3)^{24}+300(0.7)^2(0.3)^{23}+2300(0.7)^3(0.3)^{22}+ 12650(0.7)^4(0.3)^{21}+53130(0.7)^5(0.3)^{20}+177100(0.7)^6(0.3)^{19}+480700(0.7)^7(0.3)^{`18}+1081575(0.7)^8(0.3)^{17}+2042975(0.7)^9(0.3)^{16}+3268760(0.7)^{10}(0.3)^{15}\\\\=0.00177840487034\approx0.0018[/tex]

Hence, the probability that at most 10 of them will be living with both of their parents is 0.0018.

Answer: 0.4772

Step-by-step explanation:

Given : The distribution is bell shaped , then the distribution must be normal distribution.

Mean : [tex]\mu=\$280[/tex]

Standard deviation :[tex]\sigma= \$40[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x = $200

[tex]z=\dfrac{200-280}{40}=-2[/tex]

For x = $280

[tex]z=\dfrac{280-280}{40}=0[/tex]

The p-value = [tex]P(-2<z<0)=P(z<0)-P(z<-2)[/tex]

[tex]0.5-0.0227501=0.4772499\approx0.4772[/tex]

Hence, the proportion of students will spend between $200 and $280 this semester = 0.4772

Approximately 34% of the college students will spend between $200 and $280 on book purchases for a semester according to the empirical rule for a bell-shaped and symmetric distribution.

The distribution of the amount of money spent on book purchases for a semester by college students is described as bell-shaped and symmetric with a mean of $280 and a standard deviation of $40. To find the proportion of students that will spend between $200 and $280, we can use the empirical rule (68-95-99.7 rule) which indicates that approximately 68% of the data fall within one standard deviation from the mean. Since we are concerned with the range from $200 (which is 2 standard deviations below the mean) to $280 (the mean), we are effectively looking at half of this 68% range. Therefore, approximately 34% of the students are expected to spend between $200 and $280 on books for a semester.

To calculate the proportion, we take 68% of the range (which covers -1 to +1 standard deviation from the mean) and divide it by 2:

Thus, the proportion of students spending between $200 and $280 is 0.34, which when rounded to four decimal places, gives us 0.3400.

Answer: [tex]x+2y-7=0[/tex]

Step-by-step explanation:

We know that the equation of a line passing through points (a,b) and (c,d) is given by :-

[tex](y-b)=\dfrac{d-b}{c-a}(x-a)[/tex]

Then , the equation of a line passing through points (3,2) and (1,3) is given by :-

[tex](y-2)=\dfrac{3-2}{1-3}(x-3)\\\\\Rightarrow\ (y-2)=\dfrac{1}{-2}(x-3)\\\\\Rightarrow\ -2(y-2)=(x-3)\\\\\Rightarrow\ -2y+4=x-3\\\\\Rightarrow\ x+2y-7=0[/tex]

Hence, the equation of a line passing through points (3,2) and (1,3) is : [tex]x+2y-7=0[/tex]

Answer:

The current yield is 3.11%.

Step-by-step explanation:

Given - A $500 Treasury bond with a coupon rate of 2.8% that has a market value of $450.

Now the face value of the bond is = $500

The rate of interest is = 2.8%

Then interest on $500 becomes:

[tex]0.028\times500=14[/tex] dollars

The current market value is $450

So, current yield is = [tex]\frac{14}{450}\times100= 3.11[/tex]%

The current yield is 3.11%.

Final answer:

The current yield on a $500 Treasury bond with a 2.8% coupon rate and a market value of $450 is 3.11%, calculated by dividing the annual coupon payment by the market value of the bond and then converting to a percentage.

Explanation:

To calculate the current yield on the described $500 Treasury bond with a coupon rate of 2.8% that has a market value of $450, follow these steps:

Therefore, the current yield on the bond is 3.11% when rounded to two decimal places.

Answer:

[tex]30=2\: *3\:*5[/tex]

Step-by-step explanation:

We analyze between which prime numbers we can divide the number 30. The smallest prime number by which we divide is 2. Then:

[tex]\frac{30}{2}=15[/tex]

We now look for the smallest prime number that divides the 15. Since 15 is not a multiple of 2, we make the division with the number 3 that is divisor of 15.

[tex]\frac{15}{3}=5[/tex]

We now look for a number that divides to 5, but since 5 is a prime number, the only divisor other than 1 is 5. Then:

[tex]\frac{5}{5}=1[/tex]

This ends the decomposition of 30 and we find 3 prime factors:

2,3 and 5.

Answer: [tex](2.20,\ 5.26)[/tex]

Step-by-step explanation:

The confidence interval for the standard deviation is given by :-

[tex]\left ( \sqrt{\dfrac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\ \sqrt{\dfrac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}} \ \right )[/tex]

Given : n= 12 ; s= 3.1

Significance level : [tex]\alpha=1-0.95=0.05[/tex]

Using Chi-square distribution table ,

[tex]\chi^2_{11,0.025}}=21.92\\\\\chi^2_{11,0.975}}=3.82[/tex]

Now, the 95% confidence interval for the standard deviation of the height of students at UH is given by :-

[tex]\left ( \sqrt{\dfrac{(11)(3.1)^2}{21.92}},\ \sqrt{\dfrac{(11)(3.1)^2}{3.82}} \ \right )\\\\=\left ( 2.19602743525, 5.26049188471\right )\approx(2.20,\ 5.26)[/tex]

To construct a 95% confidence interval for the standard deviation of the height of students at UH, use the chi-square distribution.

To construct a 95% confidence interval for the standard deviation of the height of students at UH, we can use the chi-square distribution. Since the population standard deviation is unknown, we need to use the sample standard deviation as an estimate. The formula to calculate the confidence interval is:

Lower limit = ((n-1) * s^2) / X^2

Upper limit = ((n-1) * s^2) / X^2

Where n is the sample size, s is the sample standard deviation, and X^2 is the chi-square critical value corresponding to the desired confidence level and degrees of freedom (n-1).

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Answer:

A : 52-2

B : 0!

C : 312

D : 140608

Step-by-step explanation:

Part A:

It is given that that cards are selecting from a deck without replacing. So,

The number of ways to draw first card = 52

Now, one card is draw. The remaining cards are 51.

The number of ways to drawn second card = 52 - 1 =51

Now, one more card is drawn. The remaining cards are 50.

The number of ways to drawn third card = 52 - 2 =50

Therefore the number of ways to draw the 3 card is 52-2.

Part B:

Let a word has n letters and we need to find the number of permutations of all letters in a word, then the permutation formula is

[tex]^nP_n=\frac{n!}{(n-n)!}=\frac{n!}{0!}[/tex]

The denominator of the calculation is 0!.

Part C:

Total number of cards is a normal deck = 52

Total number sides in a die = 6

Total number of ways to draw a card from a normal deck and roll a number on a normal die = 52 × 6 = 312.

Therefore the total number of ways to draw a card from a normal deck and roll a number on a normal die is 312.

Part D:

It is given that we pick cards from a normal deck of cards, one at a time and replace the card and reshuffle the deck between draws.

Total number of ways to select each card = 52

Total number of ways to select 3 cards = 52³ = 140608

Therefore the total number of ways to select 3 cards is 140608.

Answer:

[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]

Step-by-step explanation:

We are given that  a general solution of 6th order homogeneous linear equation

[tex] y=C_1+C_2e^{-t}+C_3 Cos(3t)+C_4 Sin(3t)+C_5 tCos(3t) +C_6 sin t (3t)[/tex]

We have to find the 6th order homogeneous linear differential equation whose general solution is given above.

We know that imaginary roots are in pair

There two values of imaginary roots and the values of imaginary roots are repeat.

From first term of general solution we get D=0

From second term of general solution we get D=-1

Last four terms are the values of imaginary roots and roots are repeated.

Therefore, D=[tex]\pm 3i[/tex] and D=[tex]\pm 3i[/tex]

Substitute all values then we get

[tex]D(D+1)(D^2+9)^2=0[/tex]

[tex]D(D+1)(D^4+18D^2+81)=0[/tex]

[tex]D^6+D^5+18D^4+18D^3+81 D^2+81 D=0[/tex]

[tex](D^6+D^5+18D^4+18D^3+81 D^2+81 D)y=0[/tex]

[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]

Therefore, the 6th order homogeneous linear differential equation is

[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]

Notice that

[tex](2x^3-xy^2-2y+3)_y=-2xy-2[/tex]

[tex](-x^2y-2x)_x=-2xy-2[/tex]

so the ODE is exact, and we can find a solution [tex]F(x,y)=C[/tex] such that

[tex]F_x=2x^3-xy^2-2y+3[/tex]

[tex]F_y=-x^2y-2x[/tex]

Integrating both sides of the first equation wrt [tex]x[/tex] gives

[tex]F(x,y)=\dfrac{x^4}2-\dfrac{x^2y^2}2-2xy+3x+g(y)[/tex]

Differentiating wrt [tex]y[/tex] gives

[tex]F_y=-x^2y-2x=-x^2y-2x+g'(y)\implies g'(y)=0\implies g(y)=C[/tex]

So we have

[tex]\boxed{F(x,y)=\dfrac{x^4}2-\dfrac{x^2y^2}2-2xy+3x=C}[/tex]

The solution to this complex differential equation requires knowledge in calculus and differential equations. Without additional context, it's impossible to provide a specific solution. However, exploring technique utilization such as exact differential equations, integrating factors and substitution would be beneficial.

To solve the given differential equation, which is (2x^3 - xy^2 - 2y + 3)dx - (x^2y + 2x)dy = 0, we will use an approach of factorization or grouping like terms to simplify the equation. In some cases, you might need to rearrange terms and identify if it's a special type of differential equation, like exact, separable, or homogeneous, and then apply the relevant techniques accordingly.

This difficult task requires excellent knowledge in calculus and differential equations. Unfortunately, due to the complexity of this particular equation, without additional context or information, it is impossible to provide a specific solution. I would recommend you to go through topics such as exact differential equations, as well as methods involving integrating factors and substitution. These may help you to analyze and solve this complex equation.

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