Use generating functions to determine the number of different ways 15 identical stuffed animals can be given to six children so that each child receives at least one but no more than three stuffed animals.
Answers
Answer:
50
Step-by-step explanation:
There are 434 different ways to distribute 15 identical stuffed animals among six children so that each child receives at least one but no more than three stuffed animals.
1. Define the generating function for distributing stuffed animals to each child as:
[tex]\[ (x + x^2 + x^3)^6 \][/tex]
This represents the options for each child receiving 1, 2, or 3 stuffed animals, and there are 6 children.
2. Expand the generating function using the binomial theorem:
[tex]\[ (x + x^2 + x^3)^6 = \binom{6}{0}x^0 + \binom{6}{1}x^1 + \binom{6}{2}x^2 + \binom{6}{3}x^3 + \binom{6}{4}x^4 + \binom{6}{5}x^5 + \binom{6}{6}x^6 \][/tex]
Simplify to:
[tex]\[ 1 + 6x + 21x^2 + 56x^3 + 126x^4 + 252x^5 + 462x^6 \][/tex]
3. The coefficient of [tex]\( x^{15} \)[/tex] in the expansion represents the number of ways to distribute 15 stuffed animals among six children.
4. Identify the terms that contribute to [tex]\( x^{15} \)[/tex]:
[tex]\[ 56x^3 + 126x^4 + 252x^5 \][/tex]
5. Add the coefficients:
56 + 126 + 252 = 434
Therefore, there are 434 different ways to distribute 15 identical stuffed animals among six children so that each child receives at least one but no more than three stuffed animals.
Related Questions
blake buys paperback from the used bookstore for $ 5 . 5 . Natalie purchased an annual membership to the same bookstore for $ 35 35 so she can buy paperbacks at a discounted price $ 2.50 2.50 each. How many books would Natalie and Blake have to buy this year for their spending at the bookstore to be the same? What would their total cost be? Evaluate
Answers
Answer:
Step-by-step explanation:
1:22
Answer:
Step-by-step explanation:
so
b buys a paperback for 5.5, while after natalie purchases 35.35 for membership- her books are 2.50
blake paid 5.5
nat paid 37.85
lets see how many books blake has to pay to reach 37.85 shall we?
5.5*22=121
2.5*35=87.5
35.35+87.5=122.85
i dont think its possible
Samantha had $620 in her savings. She wanted to have at least $200 in her account after her five days in San Diego. Write an inequality to show how much she can spend each day. PLZ HELP
Answers
Answer:
Step-by-step explanation:
Samantha had $620 in her savings. She wanted to have at least $200 in her account after her five days in San Diego. This means that the amount that she can spend in 5 days would be her savings minus the least amount that she wants to have left. It becomes 620 - 200 = $420
If she decides to spend her spendable amount of $420 equally per day, it means that each day, she will spend 420/5 = $84
The inequality representing the amount that she can spend for each day will be
Let y = the amount that she can pend each day. Then, it will be
y lesser than or equal to 84
Solve the equation h = -16t2 + 255 for t, using the quadratic formula to determine the time it takes the rock to reach the canyon floor.
Answers
Answer:
t=4 seconds to reach the cannon floor
Step-by-step explanation:
ax^2+bx+c=0
0=-16t^2+255
-255=-16t^2
t^2=15.9375
t= 3.99
round it up...
t=4 seconds to reach the cannon floor
hope this helps!!! :)
Find an equation of the line that passes through the point (-1, 7) and is parallel to the line passing through the points (-3, -4) and (1, 4). (Let x be the independent variable and y be the dependent variable.)
Answers
To find a line parallel to another, determine the slope of the original line, which is 2 in this case, and then use the point-slope form with a given point and the same slope to find the equation, resulting in y = 2x + 9.
Explanation:To find an equation of the line that is parallel to another, we must first determine the slope of the given line. The line passing through the points (-3, -4) and (1, 4) has a slope calculated by the formula ∆y/∆x = (4 - (-4))/(1 - (-3)) = 8/4 = 2.
Since parallel lines have the same slope, our new line will also have a slope of 2. We can use the point-slope form of a line's equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Substituting our known point (-1, 7) and the slope 2, we get y - 7 = 2(x - (-1)). Simplified, the equation of our parallel line is y = 2x + 9.
An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population has only this risk factor (and no others). For any two of the three factors, the probability is 0.12 that she has exactly these two risk factors (but not the other). The probability that a woman has all three risk factors, given that she has A and B, is 1/3. Calculate the probability that a woman has none of the three risk factors, given that she does not have risk factor A?
Answers
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The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is calculated to be 0.54.
Explanation:To find the probability that a woman has none of the three risk factors, given that she does not have risk factor A, we will use the given probabilities and apply the principles of probability. Let's denote the probabilities of having only the risk factors as P(A), P(B), and P(C), the probabilities of having exactly two risk factors as P(A and B), P(A and C), and P(B and C), and the probability of having all three risk factors as P(A and B and C).
Given:
P(A) = P(B) = P(C) = 0.1
P(A and B) = P(A and C) = P(B and C) = 0.12
P(A and B and C | A and B) = 1/3
We can calculate P(A and B and C) using the conditional probability:
P(A and B and C) = P(A and B) × P(A and B and C | A and B) = 0.12 × 1/3 = 0.04
To find the probability of not having A, denoted as P(A'), we can use the complement rule:
P(A') = 1 - P(A) - P(A and B) - P(A and C) - P(A and B and C) = 1 - 0.1 - 0.12 - 0.12 - 0.04 = 0.62
Since P(A') includes probabilities of women with neither of the risk factors or only with B or C, we need to subtract the probabilities of having only risk factors B and C:
P(None | A') = P(A') - P(B) - P(C) + P(B and C) = 0.62 - 0.1 - 0.1 + 0.12 = 0.54
The probability that a woman has none of the three risk factors, given that she does not have risk factor A, is thus 0.54.
A rectangular poster is to contain 81 square inches of print. The margins at the top and bottom and on each side are to be 5 inches. Find the dimensions of the page which will minimize the amount of paper used.
Answers
To minimize the paper used for a poster with a specific print area and margin size, we derive a formula for total paper area, take its derivative with respect to the print width, solve for the width, and then find the matching height.
To find the dimensions of the page that will minimize the amount of paper used for a rectangular poster that contains 81 square inches of print with 5-inch margins on all sides, we need to set up a function to minimize. Let the width of the print area be x inches, and the height be y inches. Therefore, the total dimensions of the poster will be (x + 10) inches wide and (y + 10) inches high due to the margins on each side.
The area of print is given, so x*y = 81. We will minimize the total area of the page, A = (x+10)(y+10). Substituting the value of y from the print area equation, y =[tex]\frac{81}{x}[/tex], we get A(x) = (x+10)([tex]\frac{81}{x}[/tex]+10).
Now, to find the dimensions that minimize the paper used, we will take the derivative of A(x) with respect to x, set it to zero, and solve for x. From there, we can find the corresponding value of y to get the dimensions that will use the least amount of paper while still fitting the print area and margins.
What is the solution to the following system of equations?
x − 4y = 6
2x + 2y = 12
answer choices
(0,10)
(10,0)
(6,0)
(0,6)
Answers
Answer:
The answer to your question is (6, 0)
Step-by-step explanation:
Solve the system of equations by elimination
x - 4y = 6 (I)
2x + 2y = 12 (II)
Multiply (II) by 2
x - 4y = 6
4x + 4y = 24
Simplify
5x + 0 = 30
Find x
5x = 30
x = 30/ 5
x = 6
Find "y"
6 - 4y = 6
-4y = 6 - 6
-4y = 0
y = 0/-4
y = 0
Answer:
(6,0)
Step-by-step explanation:
Given equations are:
\[x - 4y = 6\] -------------------- (1)
\[2x + 2y = 12\] -------------------- (2)
Multiplying (1) by 2 :
\[2x - 8y = 12\] -------------------- (3)
Calculating (2) - (3) :
\[2x + 2y -2x + 8y = 12 - 12\]
=> \[10y =0\]
=> \[y = 0\]
Substituting the value of y in (1):
\[ x = 6 \]
So the required solution of the system of equations is x=6,y=0. This can be alternatively expressed in coordinate notation as (6,0).
A phone store employee earns a salary of $450 per week plus 10% comission on her weekly sales. A) What function rule models the employee's weekly earnings? B)If the employee earned $570 in a week, what was the amount of her sales for that week?
Answers
Answer:
A) [tex]E(w)=450+0.10w[/tex]
B) $1200
Step-by-step explanation:
Let w represent employee's weekly sales and [tex]E(w)[/tex] be total weekly earnings.
We have been given that a phone store employee earns a salary of $450 per week plus 10% commission on her weekly sales.
A) The total weekly earnings of employee would be weekly salary plus 10% of weekly sales.
10% of weekly sales, w, would be [tex]\frac{10}{100}*w=0.10w[/tex]
[tex]E(w)=450+0.10w[/tex]
Therefore, the function [tex]E(w)=450+0.10w[/tex] models the employee's weekly earnings.
B) To find the weekly sales in the week, when employee earned $570, we will substitute [tex]E(w)=570[/tex] in our formula and solve for w as:
[tex]570=450+0.10w[/tex]
[tex]570-450=450-450+0.10w[/tex]
[tex]120=0.10w[/tex]
[tex]0.10w=120[/tex]
[tex]\frac{0.10w}{0.10}=\frac{120}{0.10}[/tex]
[tex]x=1200[/tex]
Therefore, the amount of employee's weekly sales was $1200.
QUESTION 1 A researcher compares differences in positivity between participants in a low-, middle-, or upper-middle-class family. If she observes 15 participants in each group, then what are the degrees of freedom for the one-way between-subjects ANOVA?
Answers
Answer:
The degrees of freedom for the denominator on this case is given by [tex]df_{den}=df_{between}=N-K=3*15-3=42[/tex].
Step-by-step explanation:
Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".
The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"
If we assume that we have [tex]3[/tex] groups and on each group from [tex]j=1,\dots,15[/tex] we have [tex]15[/tex] individuals on each group we can define the following formulas of variation:
[tex]SS_{total}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x)^2 [/tex]
[tex]SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2 [/tex]
[tex]SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2 [/tex]
And we have this property
[tex]SST=SS_{between}+SS_{within}[/tex]
The degrees of freedom for the numerator on this case is given by [tex]df_{num}=df_{within}=k-1=3-1=2[/tex] where k =3 represent the number of groups.
The degrees of freedom for the denominator on this case is given by [tex]df_{den}=df_{between}=N-K=3*15-3=42[/tex].
And the total degrees of freedom would be [tex]df=N-1=3*15 -1 =44[/tex]
And the F statistic to compare the means would have 2 degrees of freedom on the numerator and 42 for the denominator.
A rectangular board has an area of 648 square centimeters. The triangular part of the board has an area of 162 square centimeters. A dart is randomly thrown at the board. A triangle is inside of a rectangle. The height of the triangle is the same as the height of the rectangle. Assuming the dart lands in the rectangle, what is the probability that it lands inside the triangle?
Answers
Answer:
25%.
Step-by-step explanation:
Let E be the event that the dart lands inside the triangle.
We have been given that a rectangular board has an area of 648 square centimeters. The triangular part of the board has an area of 162 square centimeters.
We know that probability of an event represents the chance that an event will happen.
[tex]\text{Probability}=\frac{\text{Favorable no. of events}}{\text{Total number of possible outcomes}}[/tex]
[tex]\text{Probability that dart lands inside the triangle}=\frac{\text{Area of triangle}}{\text{Area of rectangle}}[/tex]
[tex]\text{Probability that dart lands inside the triangle}=\frac{162}{648}[/tex]
[tex]\text{Probability that dart lands inside the triangle}=0.25[/tex]
Convert into percentage:
[tex]0.25\times 100\%=25\%[/tex]
Therefore, the probability that dart lands inside the triangle is 25%.
Answer:
25%
Step-by-step explanation:
Derrick adds equations A and B to solve this system of equations. What makes this approach a valid method in general for solving a system of equations?
Answers
Answer:
The substitution and addition properties of equality.
Step-by-step explanation:
The substitution property tells you that equals may be substituted for each other at any time.
The addition property of equality tells you that the same quantity can be added to both sides of an equation without violating the equal sign.
So, if you start with the equation ...
a = b
and you add c to both sides (addition property), you get
a + c = b + c
and if c = d, this can become (substitution property) ...
a + c = b + d . . . . . d substituted for c
In other words, we have added the equations
a = b
c = d
to get ...
a + c = b + d
The addition and substitution properties of equality make this valid.
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Suppose Adam wants to have $750,000 in his IRA at the end of 30 years. He decides to invest in an annuity paying 6% interest, compounded annually. What does he have to contribute each year to reach this goal?
Answers
Answer:
$9486.68
Step-by-step explanation:
The future value of a annuity formula can be used:
FV = P((1+r)^n -1)/r
750000 = P(1.06^30 -1)/0.06
P = 750000(.06)/(1.06^30 -1) = 9486.68
Adam has to contribute $9,486.68 each year to reach his goal.
Adam can calculate how much he needs to contribute to his IRA each year using the future value of an annuity formula. By substituting the known values into the rearranged formula, he can find the required annual payment.
Explanation:To find out how much Adam needs to contribute each year, we can use the formula for the future value of an annuity. The future value (FV) of an annuity formula is:
FV=P*[((1+r)^n -1)/r]
Where:
P is the annual payment, r is the annual interest rate (expressed as a decimal), n is the number of periods, which in this case will be years.
In this question, we want to find P, so we rearrange the formula as follows:
P = FV / [((1+r)^n -1)/r]
Substituting the given values:
P= $750,000 / [((1+0.06)^30 -1)/0.06]
By calculating the above expression, we will get the amount Adam needs to contribute annually to reach his goal.
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Find the area. The figure is not drawn to scale.
Answers
Answer:
15
Step-by-step explanation:
to find area you need to do height times base and divide by 2 or multiply it by 1/2.
10 x 3 = 30
30/2 = 15
At the center of espionage in Kznatropsk one is thinking of a new method for sending Morse telegrams. Instead of using the traditional method, that is, to send letters in groups of 5 according to a Poisson process with intensity 1, one might send them one by one according to a Poisson process with intensity 5. Before deciding which method to use one would like to know the following: What is the probability that it takes less time to send one group of 5 letters the traditional way than to send 5 letters the new way (the actual transmission time can be neglected).
Answers
Answer:
It takes less time sending 5 letters the traditional way with a probability of 36.7%.
Step-by-step explanation:
First we must take into account that:
- The traditional method is distributed X ~ Poisson(L = 1)
- The new method is distributed X ~ Poisson(L = 5)
[tex]P(X=x)=\frac{L^{x}e^{-L}}{x!}[/tex]
Where L is the intensity in which the events happen in a time unit and x is the number of events.
To solve the problem we must calculate the probability of events (to send 5 letters) in a unit of time for both methods, so:
- For the traditional method:
[tex]P(X=5)=\frac{1^{5}e^{-1}}{1!}\\\\P(X=5) = 0.367[/tex]
- For the new method:
[tex]P(X=5)=\frac{5^{5}e^{-5}}{5!}\\\\P(X=5) = 0.175[/tex]
According to this calculations we have a higher probability of sending 5 letters with the traditional method in a unit of time, that is 36.7%. Whereas sending 5 letters with the new method is less probable in a unit of time. In other words, we have more events per unit of time with the traditional method.
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1. Write an inequality for the range of the third side of a triangle if two sides measure 4 and 13.
2. If LM = 12 and NL = 7 of ∆LMN, write an inequalty to describe the lenght of MN.
3. Use the Hinge Theorem to compare the measures of AD and BD.
Answers
Answer:
1. 9 < s < 17
2. 5 < MN < 19
3. AD > BD
Step-by-step explanation:
1. The triangle inequality tells you the sum of any two sides of a triangle must exceed the length of the other side. (Some versions say, "must be not less than ..." rather than "must exceed.") In practice, this means two things:
the sum of the shortest two sides is greater than the length of the longest sidethe length of any side lies between the sum and the difference of the other two sidesHere, we can use the latter fact to write the desired inequality. The difference of the given sides is 13 -4 = 9; their sum is 13 +4 = 17. The third side must lie between 9 and 17. If that side length is designated "s", then ...
9 < s < 17
(If you don't mind a "triangle" that looks like a line segment, you can use ≤ instead of <.)
__
2. Same as (1) using different numbers.
12 -7 < MN < 12 +7
5 < MN < 19
__
3. Side CD is congruent to itself, and side CA is shown congruent to side CB. This means the requirements of the Hinge Theorem are met. That theorem tells you the longer side is opposite the greater angle:
AD > BD
Which expression is equal to 2x/x−2−x+5/x+3 ?
A. x^2+11x−6/(x−2)(x+3)
B. x^2+9x+6/(x−2)(x+3)
C. 3x^2+11x+6/(x−2)(x+3)
D. x^2+3x+10/(x−2)(x+3) I think is the correct answer.
Please help, thanks!
Answers
Answer:
D is correct
Step-by-step explanation:
2x/(x-2) - (x+5)/(x+3)
2x(x+3)/(x-2)(x+3) - (x+5)(x-2)/(x+3)(x-2)
2x^2+6x/(x-2)(x+3)-(x^2+3x-10)(x+3)(x-2)
(x^2+3x+10)/(x+3)(x-2)
Answer:
x^2+3x+10/(x-2)(x+3)
Step-by-step explanation:
Just took the test
A farmer packed 3 pints of strawberries every 4 minutes. In the afternoon she packed 2 pints of strawberries every 3 minutes. What was the difference between her morning and afternoon packing rates in pints per hour?
Answers
Answer:
5 (strawberries / hours)
Step-by-step explanation:
calculation fro morning
strawberries / minutes x minutes / hours = strawberries / hours
so after adding the value in above equation
3/4* 60/1 = 45 strawberries / hours
calculation in the afternoon
strawberries / minutes x minutes / hours = strawberries / hours
2/3 x 60/1 = 40 strawberries / hours
so now by calculating difference between morning and afternoon packing rates, you can easily calculate
45-40 = 5 (strawberries / hours)
Bowling cost $2 to rent shoes,plus $5 per game. Mini golf cost $5 to rent a club, plus $4 per game. How many games would be the same total cost for bowling and mini golf? And what is that cost
Answers
Answer:
Step-by-step explanation:
Assuming the same number of bowling and mini golf games are played, let x represent the total number of games played, either bowling or mini golf. let y represent the total cost of bowling. Let z represent the total cost of golfing
Bowling cost $2 to rent a club plus $5 per game. It means that the cost, y for x bowling games will be
y = 2 + 5x
Mini golf cost $5 to rent a club, plus $4 per game. It means that the cost, y for x mini golf games will be
z = 5 + 4x
For the total cost to be the same, we will equate both equations(y = zl
2 + 5x = 5 + 4x
5x - 4x = 5 - 2
x = 3
There would be 3 games before total cost would be the same
A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 30000 feet. The pilot would like to estimate her distance from Gateway Arch. She finds that the angle of depression to a point on the ground at the arch is 23°. Find the distance between the plane and the arch. Round your answer to the nearest foot. (Do not include ft in your answer)
Answers
Answer: 70676
Step-by-step explanation:
First we draw a diagram representing the problem, which can be found in the picture uploaded,
Point a is the point of the plane, you can see where the angle of depression is imputed in the diagram, point C is the point where the gateway arch is, and drawing a vertical line to the ground from the point of the plane, point Blank is where that vertical line touches the ground
So we can tell the the angle of depression from the plane to the arch is the same as the angel of elevation from the arch to the plane
And we are to look for the distance between B and C which is labeled x in the diagram
So looking at the right angle triangle made from this question, we can see we have the opposite length which the angle of elevation from the arch is looking at, and we are looking for the adjacent length, so we use SOH, CAH, TOA, to solve
Choosing TOA which means
Tan(angle) = (opposite length)/(adjacent length)
Tan 23 = 30000/x
Multiplying both sides by x
xtan23 = 30000
Dividing both sides by tan23
x = 30000/tan23
x = 70675.57
Approximately 70676
Question 1(Multiple Choice Worth 1 points) (01.04 LC) Solve for x: −3x + 3 < 6 x > −1 x < −1 x < −3 x > −3
Answers
To solve -3x + 3 < 6 and find the value of x, subtract 3 from both sides and divide by -3. The solution is x > -1.
Explanation:To solve the inequality -3x + 3 < 6, we want to isolate the variable x. First, subtract 3 from both sides to get -3x < 3. Then, divide both sides by -3 (remember to flip the inequality sign when dividing by a negative) to obtain x > -1. Therefore, the solution is x > -1.
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The half life of a certain tranquilizer in the bloodstream is 37 hours. How long will it take for the drug to decay to 86% of the original decay model,A=A
Answers
Answer:
8.1 hours
Step-by-step explanation:
A model of the fraction remaining can be ...
f = (1/2)^(t/37) . . . . t in hours
So, for the fraction remaining being 86%, we can solve for t using ...
0.86 = 0.5^(t/37)
log(0.86) = (t/37)log(0.5)
t = 37·log(0.86)/log(0.5) ≈ 8.0509 ≈ 8.1 . . . hours
It takes about 8.1 hours to decay to 86% of the original concentration.
A study found out that 1% of social security recipients are too young to vote. If 800 social security recipients are randomly selected, find the mean, variance and standard deviation of the number of recipients who are too young to vote
Answers
Answer:
Mean : [tex]\mu=8[/tex]
Variance : [tex]\sigma^2=7.92[/tex]
Standard deviation = [tex]\sigma=2.81[/tex]
Step-by-step explanation:
We know that , in Binary Distribution having parameters p (probability of getting success in each trial) and n (Total number trials) , the mean and variance is given by:-
Mean : [tex]\mu=np[/tex]
Variance : [tex]\sigma^2=np(1-p)[/tex]
We are given that ,
Total social security recipients : n=800
The probability of social security recipients are too young to vote : p=1%= 0.01
Here success is getting social security recipients are too young to vote .
Then, the mean, variance and standard deviation of the number of recipients who are too young to vote will be :-
Mean : [tex]\mu=800\times0.01=8[/tex]
Variance : [tex]\sigma^2=800\times 0.01(1-0.01)=8\times0.99=7.92[/tex]
Standard deviation = [tex]\sigma=\sqrt{\sigma^2}=\sqrt{7.92}=2.81424945589\approx2.81[/tex]
Hence, the mean, variance and standard deviation of the number of recipients who are too young to vote :
Mean : [tex]\mu=8[/tex]
Variance : [tex]\sigma^2=7.92[/tex]
Standard deviation = [tex]\sigma=2.81[/tex]
In a sample of 800 social security recipients, with 1% being too young to vote, the mean is 8, the variance is 7.92, and the standard deviation is approximately 2.81.
The study indicates that 1% of social security recipients are too young to vote. When sampling 800 social security recipients, we treat the number of recipients too young to vote as a binomial random variable (since each recipient is either too young or not, with a fixed probability of being too young).
To find the mean of the binomial distribution, we use the formula:
Mean = n * p
Where n is the sample size (800) and p is the probability of success (0.01).
Mean = 800 * 0.01 = 8
The variance of the binomial distribution is given by the formula:
Variance = n * p * (1 - p)
Variance = 800 * 0.01 * (1 - 0.01) = 7.92
To calculate the standard deviation, we take the square root of the variance.
Standard Deviation = √(Variance) = √(7.92) = 2.81
What is the sum of a common geometric series if the first term is 8 and the common ratio is 1/2?
Answers
Answer: A
Step-by-step explanation:
The sum to infinity of a geometric series is
S (∞ ) = \frac{a}{1-r} ( - 1 < r < 1 )
where a is the first term 8 and r is the common ratio, hence
S(∞ ) = {8}{1-\{1}{2} } = {8}{1}{2} } = 16
Answer:
Step-by-step explanation:
32 i think
A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses, estimate the probability of getting at least 20% correct. 0.1492 0.3508 0.0901 0.8508 Normal approximation is not suitable.
Answers
Answer:
Option 4 - 0.8508
Step-by-step explanation:
Given : A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses.
To find : Estimate the probability of getting at least 20% correct ?
Solution :
20% correct out of 60,
i.e. [tex]20\%\times 60=\frac{20}{100}\times 60=12[/tex]
Minimum of 12 correct out of 60 i.e. x=12
Each question has 4 possible answers of which one is correct.
i.e. probability of answering question correctly is [tex]p=\frac{1}{4}=0.25[/tex]
Total question n=60.
Using a binomial distribution,
[tex]P(X\geq 12)=1-P(X\leq 11)[/tex]
[tex]P(X\geq 12)=1-[P(X=0)+P(X=1)+P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)][/tex]
[tex]P(X\geq 12)=1- [^{60}C_0(0.25)^0(1-0.25)+^{60-0}+^{60}C_1(0.25)^1(1-0.25)^{60-1}+^{60-1}+^{60}C_2(0.25)^2(1-0.25)^{60-2}+^{60}C_3(0.25)^3(1-0.25)^{60-3}+^{60}C_4(0.25)^4(1-0.25)^{60-4}+^{60}C_5(0.25)^5(1-0.25)^{60-5}+^{60}C_6(0.25)^6(1-0.25)^{60-6}+^{60}C_7(0.25)^7(1-0.25)^{60-7}+^{60}C_8(0.25)^8(1-0.25)^{60-8}+^{60}C_9(0.25)^9(1-0.25)^{60-9}+^{60}C_{10}(0.25)^{10}(1-0.25)^{60-10}+^{60}C_{11}(0.25)^{11}(1-0.25)^{60-11}][/tex]
[tex]P(X\geq 12)\approx 0.8508 [/tex]
Therefore, option 4 is correct.
A food manufacturer uses an extruder (a machine that makes bite size cookies)that yields revenue for the firm at a rate of $200 per hour when in operation. However, the extruder breaks down an average of two times every day it operates. If Y denotes the number of breakdowns per day, the daily revenue generated by the machine is
R=1600−50Y².
Find the expected daily revenue for the extruder.
Answers
Answer:
$1300
Step-by-step explanation:
The extruder yields a revenue of $200per hour
Y denotes the number of breakdown per day.
The daily revenue generated is given as
R = 1600 - 50Y^2
We have an average of 2 breakdown per day
Lamda = 2
Represent lamda as β
E(Y) = β
E(Y(Y-1)) = β^2
E(Y^2) = E[Y(Y-1)] + E(Y)
= β^2 + β
E(R) = E(1600 - 50Y^2)
= 1600 - 50E(Y^2)
= 1600 - 50(β^2 +β)
Recall that β = lamda = 2
= 1600 - 50(2^2 + 2)
= 1600 - 50(4+2)
= 1600 - 50(6)
= 1600 - 300
= 1300
$1300
The expected daily revenue of the extruder is $1300
Two cars leave towns 400 kilometers apart at the same time and travel toward each other. One car's rate is 14 kilometers per hour less than the other's. If they
meet in 2 hours, what is the rate of the slower car?
Do not do any rounding.
Answers
Answer:
Step-by-step explanation:
v*2+(v-14)*2=400
2v+2v-28=400
4v=400+28
4v=428
v=107 km/h
speed of slowest car=107-14=93 km/h
Joe has $200 in his savings account and is depositing $50 per month. Kathy has $50 in her account and is depositing $75 per month. In how many months will they have the same amount of money? Please show work
Answers
In 6 months, they will have same amount of money.
Step-by-step explanation:
Let,
x be the number of months
Given,
Joe's savings = $200
Per month deposit = $50
J(x)=200+50x Eqn 1
Kathy savings = $50
Per month deposit = $75
K(x)=50+75x Eqn 2
For same amount;
J(x)=K(x)
[tex]200+50x=50+75x\\50x-75x=50-200\\-25x=-150[/tex]
Dividing both sides by -25;
[tex]\frac{-25x}{-25}=\frac{-150}{-25}\\x=6[/tex]
In 6 months, they will have same amount of money.
Keywords: functions, division
Learn more about division at:
brainly.com/question/4034547brainly.com/question/4054269#LearnwithBrainly
A road that is 11 miles long is represented on a map that shows a scale of 1 centimeter being equivalent to 10 kilometers. How many centimeters long does the road appear on the map? Round your result to the nearest tenth of a centimeter.
Answers
The 11 kilometer road will be 1.1 cm long on the map
Step-by-step explanation:
Scale is used on maps to show large locations or roads as small representatives of the larger objects.
The scale factor is usually in proportion to the original length or dimensions.
Given that
1 cm = 10 km
Then, we will divide the number of kilometers by 10 to find the length of road on map
11 km on map = [tex]\frac{11}{10}[/tex]
Hence,
The 11 kilometer road will be 1.1 cm long on the map
Keywords: Maps, Scales
Learn more about maps at:
brainly.com/question/3398261brainly.com/question/3614284#LearnwithBrainly
Let H be a subgroup of a group G. We call H characteristic in G if for any automorphism σ∈Aut(G) of G, we have σ(H)=H.
(a) Prove that if σ(H)⊂H for all σ∈Aut(G), then H is characteristic in G.
(b) Prove that the center Z(G) of G is characteristic in G.
Answers
Answer:Problem 1. Let G be a group and let H, K be two subgroups of G. Dene the set HK = {hk : h ∈ H,k ∈ K}.
a) Prove that if both H and K are normal then H ∩ K is also a normal subgroup of G.
b) Prove that if H is normal then H ∩ K is a normal subgroup of K.
c) Prove that if H is normal then HK = KH and HK is a subgroup of G.
d) Prove that if both H and K are normal then HK is a normal subgroup of G.
e) What is HK when G = D16, H = {I,S}, K = {I,T2,T4,T6}? Can you give geometric description of HK?
Solution: a) We know that H ∩ K is a subgroup (Problem 3a) of homework 33). In order to prove that it is a normal subgroup let g ∈ G and h ∈ H ∩ K. Thus h ∈ H and h ∈ K. Since both H and K are normal, we have ghg−1 ∈ H and ghg−1 ∈ K. Consequently, ghg−1 ∈ H ∩ K, which proves that H ∩ K is a normal subgroup.
b) Suppose that H G. Let K ∈ k and h ∈ H ∩ K. Then khk−1 ∈ H (since H is normal in G) and khk−1 ∈ K (since both h and k are in K), so khk−1 ∈ H ∩ K. This proves that H ∩ K K.
c) Let x ∈ HK. Then x = hk for some h ∈ H and k ∈ K. Note that x = hk = k(k−1hk). Since k ∈ K and k−1hk ∈ H (here we use the assumption that H G), we see that x ∈ KH. This shows that HK ⊆ KH. To see the opposite inclusion, consider y ∈ KH, so y = kh for some h ∈ H and k ∈ K. Thus y = (khk−1)k ∈ HK, which proves that KH ⊆ HK and therefoere HK = KH. To prove that HK is a subgroup note that e = e · e ∈ HK. If a,b ∈ HK then a = hk and b = h1k1 for some h,h1 ∈ H and k,k1 ∈ K. Thus ab = hkh1k1. Since HK = KH and kh1 ∈ KH, we have kh1 = h2k2 for some k2 ∈ K, h2 ∈ H. Consequently,
ab = h(kh1)k1 = h(h2k2)k1 = (hh2)(k2k1) ∈ HK
(since hh2 ∈ H and k2k1 ∈ K). Thus HK is closed under multiplication. Finally,
Step-by-step explanation:
Add or subtract
x/x^2-4 -2/x^2-4
Answers
Simplify the expression.
−8x^2-x-4/x^2
That dash is for fraction