If I had $3 in my bank account and took away $2, how many dollars would i have?

The type of sampling used in this example is simple random sampling.

Simple random sampling is a type of probability sampling where each member of the population has an equal chance of being selected. In this example, the pollster generated 500 random numbers and interviewed the corresponding voters, which means that each voter had an equal chance of being selected.

This method is often used when the population is homogeneous and there is no need to stratify or cluster the sample. It is also relatively easy to implement and has a low margin of error if the sample size is large enough.

Therefore, the type of sampling used in this example is simple random sampling.

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Answer:

To dilate by a scale factor of 2, multiply the length of each side of the shape by 2. If the shape is on a coordinate plane, multiply the coordinates of each vertex of the shape by 2.

Hope it helps

The correct description of Standard deviation is option A: the standard deviation of all samples of size.

Standard deviation is a measure of the dispersion of a set of data from its mean. It provides information about how spread out the data is from the average value. It is calculated by finding the square root of the variance, which is the average of the squared differences from the mean. Standard deviation is used in various fields like finance, physics, and statistics. It is important to note that the larger the standard deviation, the more spread out the data is and the more diverse the samples are. Therefore, it is a useful tool in understanding the variability of data.

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Answer:

D

Step-by-step explanation:

The Pythagorean theorem states that [tex]a^2+b^2=c^2[/tex], since we know the values of a and b, we can solve for c by square rooting both sides of the equation.

[tex]\sqrt{a^2+b^2}=\sqrt{c^2[/tex]

[tex]c=\sqrt{a^2+b^2[/tex]

[tex]c = \sqrt{14^2+48^2}[/tex]

[tex]c=50[/tex]

The linear equation that shows a proportional relationship is "y = −2." It exhibits a constant ratio between y and x, maintaining a proportional relationship as x varies.

A linear equation represents a proportional relationship when the ratio between the dependent variable (y) and the independent variable (x) remains constant.

Out of the given options, the equation that exhibits a proportional relationship is "y = −2". In this equation, the coefficient of x is -2, indicating that y is always equal to -2 times the value of x.

As x increases or decreases, y changes proportionally in the opposite direction, maintaining a constant ratio. For example, when x = 1, y = -2; when x = -2, y = 4, and so on. Therefore, the equation y = -2 exhibits a proportional relationship.

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Therefore, the indefinite integral of the given function is x^3 + (4/3)x^4 - (4/9)x^(9/2) + C, where C is the constant of integration.

We can begin by simplifying the integrand as follows:

x^2(3 + 4x - 4x^(3/2)) dx

= 3x^2 dx + 4x^3 dx - 4x^(7/2) dx

= x^3 + (4/3)x^4 - (4/9)x^(9/2) + C

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The ordered pair for each point using the grid is (4,3). An ordered pair is a set of two numbers that represent a point in the coordinate plane.

In general, the first number represents the horizontal or x-coordinate, and the second number represents the vertical or y-coordinate. These coordinates are usually represented in brackets and separated by a comma. For example, (2,3) is an ordered pair whose x-coordinate is 2, and whose y-coordinate is 3. What is a grid? A grid is a set of horizontal and vertical lines that are used to form boxes, creating a pattern of squares. In mathematics, we use grids to represent the coordinate plane. The x-axis, which runs horizontally, is marked with positive numbers to the right and negative numbers to the left of the origin. The y-axis, which runs vertically, is marked with positive numbers above and negative numbers below the origin. How to write the ordered pair for each point? To write the ordered pair for each point using the grid, we need to follow these steps: Step 1: Locate the point on the grid. Let's say we have a point that lies in the third row (y-coordinate) and the fourth column (x-coordinate). We would locate this point by counting three rows down and four columns to the right. This point is shown below. Step 2: Write the ordered pair for the point. In this case, the x-coordinate is 4, and the y-coordinate is 3. We write these coordinates in brackets, separated by a comma. Therefore, the ordered pair for this point is (4, 3). The ordered pair for each point using the grid is (4,3). The first number in the ordered pair is the x-coordinate, and the second number is the y-coordinate.

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The solution to the inequality is x ≥ 2, given that x is not equal to -3/2, and the solution is x ∈ [2, ∞).

To find the domain of the inequality (3x - 1)/(2x + 3) ≥ 1, we need to look for values of x that make the denominator 2x + 3 equal to zero, since division by zero is undefined. We can set the denominator equal to zero and solve for x:

2x + 3 = 0

2x = -3

x = -3/2

Therefore, x cannot equal -3/2.

To solve the inequality, we can start by multiplying both sides by 2x + 3 to clear the denominator:

[tex](3x - 1)\div(2x + 3) \times (2x + 3) \geq1 \times (2x + 3)[/tex]

3x - 1 ≥ 2x + 3

x ≥ 2

Therefore, the solution to the inequality is x ≥ 2, but we need to check that this solution is within the domain of the inequality. Since x cannot equal -3/2, the solution is:

x ∈ [-3/2, ∞) ∩ [2, ∞) = [2, ∞)

The domain of the inequality is x ∈ (-∞, -3/2) ∪ (-3/2, ∞).

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We can see here that a parabolic solar cooker is one potential gadget that brings the reflecting qualities of conic sections to a practical setting. All light rays parallel to a parabola's axis are reflected to a single point, known as the focus, making it a conic section.

A device is a tool, machine, or instrument that is designed to perform a specific function or task.

The reflecting qualities of the parabola can be utilized to focus and concentrate light energy for cooking, heating, or other uses. This gadget shows how conic sections can be employed practically in real-world settings.

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The probability that your guess is wrong is 3/8 or approximately 0.375.

To find the probability that your guess is wrong, we need to consider the different scenarios and calculate the corresponding probabilities.

Let's denote the events as follows:

A: Unit selected from Manufacturer A

B: Unit selected from Manufacturer B

D: Selected unit is defective

G: Your guess is wrong (incorrectly guessing the manufacturer)

We are given:

P(A) = P(Unit from Manufacturer A) = 1/2 (since you select a box at random)

P(D|A) = Probability of selecting a defective unit from Manufacturer A = 1/4

P(D|B) = Probability of selecting a defective unit from Manufacturer B = 1/2

We need to calculate P(G), the probability that your guess is wrong.

Using Bayes' theorem:

P(G) = P(G|A) * P(A) + P(G|B) * P(B)

We know that if a unit is defective, your guess will be that it came from Manufacturer B, and if it is not defective, your guess will be that it came from Manufacturer A.

Therefore:

P(G|A) = P(D|A)

P(G|B) = 1 - P(D|B) = 1 - 1/2 = 1/2

Plugging in the values:

P(G) = P(D|A) * P(A) + P(G|B) * P(B)

    = (1/4) * (1/2) + (1/2) * (1/2)

    = 1/8 + 1/4

    = 3/8

Therefore, the probability that your guess is wrong is 3/8 or approximately 0.375.  

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The equation graphed is  y = 0.003(x - 50)² - 7.5

The bridge sags the most 7.5 feet from the left bank

A 3-foot will be eve with the river at about 11 feet and 87 feet from the left bank

Standard vertex form, y = a(x - h)² + k    

The vertex from the graph is

v (h, k) = (50, -7.5)

y = a(x - 50)² - 7.5

solving for using (0, 0)

0 = a(0 - 50)² - 7.5

a = 7.8 / 50²

a = 0.00312

hence the equation is y = 0.003(x - 50)² - 7.5

where y = -3 from the graph the point is traced to be  approximately 11 feet and 87 feet from the left bank

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The p-value is approximately 0.0879, which is greater than the significance level of 0.05.

a. The test statistic can be calculated using the formula:

t = (d bar- μd) / (s / √n)

where d bar is the sample mean difference, μd is the hypothesized population mean difference (in this case, 0), s is the sample standard deviation of the differences, and n is the sample size.

Using the given data, we have:

d bar = (82+86+69+75+87+94)/6 - (77+82+68+71+78+85)/6 = 5.5

s = √[(82-77.5)² + (86-82.5)² + ... + (94-85)² / (6-1)] ≈ 8.572

n = 6

Plugging these values into the formula, we get:

t = (5.5 - 0) / (8.572 / √6) ≈ 2.058

Therefore, the test statistic is approximately 2.058.

b. To test whether taking the course actually increased scores on the driving exam, we will use a two-tailed t-test with a significance level of 0.05. The null hypothesis is that the population mean difference is 0, and the alternative hypothesis is that the population mean difference is not 0.

Using a t-distribution table with 5 degrees of freedom (n-1), we find that the critical values for a two-tailed test at a significance level of 0.05 are approximately ±2.571.

Since the test statistic of 2.058 falls within the acceptance region, we fail to reject the null hypothesis. This means that we do not have sufficient evidence to conclude that taking the course actually increased scores on the driving exam.

To calculate the p-value, we can use a t-distribution table or a t-distribution calculator. The p-value for a two-tailed t-test with a test statistic of 2.058 and 5 degrees of freedom is approximately 0.0879 (rounded to four decimal places).

Therefore, the p-value is approximately 0.0879, which is greater than the significance level of 0.05. This further supports the conclusion that we do not have sufficient evidence to conclude that taking the course actually increased scores on the driving exam.

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By using exponents, we can write the expressions  (6⁵)¹⁰ as  6⁵⁰.

The exponent of a number says how many times to use the number in a multiplication.

Also, exponent is defined as the method of expressing large numbers in terms of powers. That means, exponent refers to how many times a number multiplied by itself.

If  an expression is given as (6⁵)¹⁰ , using exponents, we can write the  expressions in the form of [tex]6^k[/tex] as shown below;

The expression "k" is an integer {0, 1, 2, 3, etc}

By applying rule of exponent for the multiplication of powers, (6⁵)¹⁰ is simplified as;

(6⁵)¹⁰ = 6⁵⁰

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The area of the region enclosed by one loop of the curve r = 3 cos(5θ) is (9/2) π square units

To find the area of the region enclosed by one loop of the curve given by the polar equation r = 3 cos(5θ), we can integrate over the corresponding range of θ values.

The curve r = 3 cos(5θ) represents a cardioid with five petals.

To determine the range of θ values that corresponds to one loop, we can set the equation inside the cosine function equal to zero:

5θ = 0

This gives us θ = 0.

So, for one complete loop, we need to integrate from θ = 0 to θ = 2π.

The area formula for a polar curve is given by:

A = (1/2) ∫[θ₁,θ₂] r(θ)² dθ

In this case, the area can be calculated as:

A = (1/2) ∫[0, 2π] (3 cos(5θ))² dθ

Simplifying the integral, we have:

A = (9/2) ∫[0, 2π] cos²(5θ) dθ

Using the trigonometric identity cos²(θ) = (1 + cos(2θ))/2, we can rewrite the integral:

A = (9/2) ∫[0, 2π] (1 + cos(10θ))/2 dθ

Expanding the integral, we get:

A = (9/4) ∫[0, 2π] dθ + (9/4) ∫[0, 2π] (cos(10θ))/2 dθ

The first integral ∫ dθ over the interval [0, 2π] gives us 2π:

A = (9/4) (2π) + (9/8) ∫[0, 2π] cos(10θ) dθ

The second integral ∫ cos(10θ) dθ can be evaluated as:

(1/10) sin(10θ)

Evaluating the integral over the interval [0, 2π], we get:

A = (9/4) (2π) + (9/8) [(1/10) sin(10(2π)) - (1/10) sin(10(0))]

Since sin(0) = 0 and sin(20π) = 0, the second term becomes zero:

A = (9/4) (2π) + (9/8) (0)

Simplifying, we have:

A = (9/4) (2π)
A = (9/2) π

Therefore, the area of the region enclosed by one loop of the curve r = 3 cos(5θ) is (9/2) π square units.

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Answer:

The correct answer is: y=cosx + 2

Answer: Option B

Step-by-step explanation: To analyze the given graph let us,

compare the given to a known, standard function

take maxima and minima points from the graph and analyze them.

From the given graph we can pick the points which can tell us about the nature of the cosine function we took,

It is clear from the graph that the function plotted is 2 times greater than the standard function i.e. cos(x)

in cos(x) ⇒ we have minima at (-π, 1)

in the graph, ⇒ we have minima at  ( , -1)

By a factor of 2, there is a change in the graph with respect to the standard cosine graph.

hence, the graph given is cos(2x)  

we know that,

at (0,1) the cosine graph touches the y-axis.

in the given graph, the function has a maximum and touches the y-axis at (0,4)

therefore, 4× cos(2x) is the function represented in the graph.

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We have shown that the determinant of the 3x3 matrix M, divided by 2, gives us the area of the triangle formed by the three given vertices on a coordinate plane.

To prove that the determinant of a 3x3 matrix representing three vertices of a triangle on a coordinate plane can solve the area of a triangle, we can use the following steps:

Let's consider a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3).

Step 1: Construct the matrix M:

M = | x1 y1 1 |

| x2 y2 1 |

| x3 y3 1 |

Step 2: Calculate the determinant of matrix M:

det(M) = (x1 * y2 * 1 + y1 * 1 * x3 + 1 * x2 * y3) - (1 * y2 * x3 + x1 * 1 * y3 + y1 * x2 * 1)

= (x1 * y2 + y1 * x3 + x2 * y3) - (y2 * x3 + x1 * y3 + y1 * x2)

= x1 * y2 + y1 * x3 + x2 * y3 - y2 * x3 - x1 * y3 - y1 * x2

Step 3: Simplify the determinant:

det(M) = x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2)

Step 4: Divide the determinant by 2 to get the area of the triangle:

Area = 1/2 * det(M)

= 1/2 * [x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2)]

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When a2 must be completed before a4 in a dependency matrix, the activities are said to be "dependent."

A dependency matrix is a tool used in project management to map out the relationships between different activities in a project. It is used to identify the order in which activities should be completed and to determine which activities are dependent on others. In a dependency matrix, a2 and a4 are two separate activities.

When a2 must be completed before a4, it means that the completion of a4 is dependent on the completion of a2. Therefore, a4 cannot be started until a2 has been completed. Activities that are dependent on other activities are referred to as "dependent" activities.

It is important to identify and manage dependent activities in a project to ensure that the project is completed on time and within budget.

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Here is my proposed redesign of the cereal box to reduce manufacturing costs while maintaining the same volume:

Original box dimensions:

- Surface Area: 334 in^2

- Volume: 312 in^3

Since surface area is proportional to the cost of materials and manufacturing, a design with lower surface area will cost less to produce.

Proposed new design: Cylinder

To hold the same 312 in^3 volume, the cylinder would have:

Radius: 3.68 inches

Height: 8.46 inches

Surface Area of cylinder:

2πrh + 2πr^2

= 2(3.14)(3.68)(8.46) + 2(3.14)(3.68)^2

= 300 in^2

Since the surface area of the cylindrical box is less than the original at 300 in^2 versus 334 in^2, the cylindrical design will require less material and cost less to produce while maintaining the same 312 in^3 volume to hold the same amount of cereal.

In summary, by changing from a rectangular prism shape to a cylindrical shape with a radius of 3.68 inches and height of 8.46 inches, we can reduce the surface area from 334 in^2 to 300 in^2. This lower surface area translates to lower material costs and manufacturing costs while still providing the original 312 in^3 volume capacity.

Answer:

now.. notice the picture below

it has a front-and-back of 11x4

it has a left-and-right of 6x4

and a top-and-bottom of 6x11

now, if the Surface Area is less than 290in², then it is cheaper to manufacture because it uses less material to make the box

explanation 2 If you dont understand Explanation 1

The new cereal box is in the form of a cube.

So from the formula of volume of cube a a³

Where a is the side of the cube.

This volume is equal to the volume of the original box

a³ = 264 inch³

a = 6.415 inch

Now the surface area of the box will be

S = 6a²

S = 6 × (6.415)² = 246.91 inch²

Which is less than the surface area of the original box.

Therefore, the manufacturing cost of the new cereal box will be less.

Step-by-step explanation:

The lengths of the legs of the 30-60-90 triangle with a hypotenuse of 24 inches are:

Short leg: 12 inches

Long leg: 12√3 inches

In a 30-60-90 triangle, the lengths of the sides are related by a specific ratio.

Let's denote the short leg as "x," the long leg as "y," and the hypotenuse as "24 inches."

The ratio for a 30-60-90 triangle is:

Short leg : Long leg : Hypotenuse = 1 : √3 : 2

Since the hypotenuse is given as 24 inches, we can use this ratio to find the lengths of the short and long legs.

Short leg = 1 × (24 inches / 2) = 12 inches

Long leg = √3 × (24 inches / 2) = 12√3 inches

The lengths of the sides in a triangle with sides of 30-60-90 are inversely proportional.

Assign the short leg the letter "x," the long leg the letter "y," and the hypotenuse the number "24 inches."

The triangle's ratio is 30-60-90:

brief leg lengthy leg Hypotenuse = 1, 3, and 2.

We may use this ratio to determine the lengths of the short and long legs because the hypotenuse is specified as 24 inches.

Short leg: 1 (24 inches divided by 2) equals 12 inches

Long leg = (24 inches divided by 2) / 3 = 12 3/8 inches

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The calculated value of the mean of the data values $35, $42, $52, $38, and $42 is $41.8

From the question, we have the following parameters that can be used in our computation:

Workers = 5

Hourly rates = $35, $42, $52, $38, and $42.

The mean of the data is calculated as

Mean = Sum of data/Number of workers

Substitute the known values in the above equation, so, we have the following representation

Mean = (35 + 42 + 52 + 38 + 42)/5

Evaluate the sum of the numerator

Mean = 209/5

Evaluate the quotient of 209 and 5

Mean = 41.8

Hence, the mean of the data is $41.8

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Complete question

Five plumbers charge different hourly rates: $35, $42, $52, $38, and $42.

Find the mean of the data.

The value of the derivative, f'(x), is equal to 16x + 7.

To find f'(x), we need to take the derivative of the given function. Using the power rule and the constant multiple rule, we get:
f'(x) = 16x + 7

Now that we have the derivative, we can use the given condition f(0) = 3 to solve for the constant of integration.

We know that:
f(x) = ∫ f'(x) dx

So we can integrate f'(x) to get:
f(x) = 8x² + 7x + C
where C is the constant of integration.

Using f(0) = 3, we get:
f(0) = 8(0)² + 7(0) + C = 0 + 0 + C = C = 3

So the final equation for f(x) is:
f(x) = 8x² + 7x + 3

And the derivative is:
f'(x) = 16x + 7

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Answer:

----------------------

There are four elements of data.

The median is the average of the middle two terms:

Answer:

6.5

Step-by-step explanation:

First, you should arrange the data in ascending to descending to find the median.

5, 5, 8, 15

Now let us use the given formula to find the median.

[tex]\sf \dfrac{n+1}{2} =--^t^h data[/tex]

Here,

n → the number of elements

Let us find it now.

[tex]\sf Median= \dfrac{n+1}{2}\\\\\sf Median=\dfrac{4+1}{2} =2.5^n^d data\\\\\sf Median=\dfrac{2^n^d data+3^r^d data}{2}\\\\Median=\dfrac{5+8}{2}\\\\Median=\dfrac{13}{2}\\\\Median=6.5[/tex]

If a sample of 35 such reviews is selected then approximately 38.97% of the 35 reviews will take more than an hour.

For calculating what percent of reviews will take more than an hour, we can use the standard normal distribution to answer this question.

First, we need to standardize the review time of an IRS examiner by subtracting the mean and dividing by the standard deviation:

z = (60 - 55.1) / 17.6 = 0.278

where 60 is the number of minutes in an hour.

This means that a review time of one hour (60 minutes) corresponds to a standard score of 0.278.

To find the probability that a review time is more than an hour (60 minutes), we need to find the area under the standard normal curve to the right of this z-score:

P(Z > 0.278) = 0.3897

Using this probability, we can conclude that approximately 38.97% of the 35 reviews will take more than an hour.

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The probability of rolling a 2 on a single die is 1/6, so the expected number of 2's in 100 rolls is μ = np = 100*(1/6) = 16.67. We can approximate the number of 2's in 100 rolls using a normal distribution with mean μ = 16.67 and variance σ^2 = np(1-p) = 13.89, since n is large enough and the probability of rolling a 2 on a single roll is not too small or too large.

To find the probability P(20 ≤ X ≤ 25), where X is the number of 2's rolled in 100 rolls, we can standardize X using the standard normal distribution, which gives:

z = (X - μ) / σ = (X - 16.67) / √13.89

Using a standard normal distribution table or calculator, we can find the probability that z lies between two values, such as -1.36 and -0.53, which correspond to 20 and 25, respectively. This gives:

P(-1.36 ≤ z ≤ -0.53) = Φ(-0.53) - Φ(-1.36) ≈ 0.137

Therefore, the probability of rolling between 20 and 25 2's in 100 rolls of a die is approximately 0.137, using the standard normal distribution as an approximation.

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Answer:

y = -3x+18

Step-by-step explanation:

We can find the slope using the formula

m = ( y2-y1)/(x2-x1)  where *x1,y1) and (x2,y2) are two points on the line.

( 1,15) and (5,3) are two points on the line.

m = ( 3-15)/(5-1)

m = -12/4

  = -3

The slope is negative.

Using the equation for a line.

y = mx+b

y = -3x+b

Substituting a point into the equation.

15 = -3(1) + b

18 = b

y = -3x+18

simultaneous equation

The probability of rolling an odd number exactly 4 times when rolling ten dice, is approximately 0.2063.

To find the probability, p(x = 4), of rolling an odd number exactly 4 times when rolling ten dice, we need to calculate the probability of getting 4 successes (rolling an odd number) and 6 failures (rolling an even number).

In this experiment, each die roll is independent and has a 50% chance of rolling an odd number (success) and a 50% chance of rolling an even number (failure). Therefore, the probability of success, p, is 0.5.

We can use the binomial probability formula to calculate p(x = 4):

[tex]p(x = 4) = C(10, 4) * p^4 * (1 - p)^{(10 - 4)[/tex]

C(10, 4) is the number of combinations of 10 dice rolls taken 4 at a time and can be calculated as:

C(10, 4) = 10! / (4! * (10 - 4)!) = 210

Substituting the values into the formula:

[tex]p(x = 4) = 210 * (0.5)^4 * (1 - 0.5)^{(10 - 4)[/tex]

= 210 * 0.0625 * 0.015625

≈ 0.2063

Therefore, p(x = 4), the probability of rolling an odd number exactly 4 times when rolling ten dice, is approximately 0.2063.

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(a) The number of different color combinations for filling up a jar with 50 jellybeans of 4 different colors, with no restrictions on the distribution of colors, is 4^50.

(b) The number of distinct non-negative integer solutions to the equation x1 +x2+x3 + x4 +x5 +x6 +x7 =20 is 230,230, which is found using the stars and bars method.

(a) Since there are 4 colors of jellybeans and no restrictions on the distribution of colors, we can choose any combination of colors to fill up the jar. Each jellybean can be one of 4 colors, so the number of different color combinations is 4^50.

(b) This is a classic stars and bars problem. We need to find the number of non-negative integer solutions to the equation x1 +x2+x3 + x4 +x5 +x6 +x7 =20. To solve this, we can imagine representing each x variable with a star ( * ), and using bars ( | ) to separate the stars into 7 groups, with each group representing one of the x variables. For example, the solution x1=2, x2=3, x3=0, x4=5, x5=4, x6=1, x7=5 would be represented as:

**|***||*****|||||||||

There are a total of 20 stars in this diagram, and 6 bars separating them into 7 groups. The number of solutions is equal to the number of ways to arrange the 20 stars and 6 bars, which is (20+6) choose 6, or 26 choose 6, which is equal to 230,230. Therefore, there are 230,230 distinct solutions to the equation x1 +x2+x3 + x4 +x5 +x6 +x7 =20.

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Your answer is: x* = Py/Px. The solution for x* in terms of Px, Py, and I is Py / Px.

Explanation:
Step 1: Given U = ln(x) + y, find the partial derivatives with respect to x and y.
∂U/∂x = 1/x
∂U/∂y = 1

Step 2: According to the theory of utility maximization, the consumer maximizes utility subject to the budget constraint Px * x + Py * y = I.

Step 3: Set up the Lagrangian function L(x, y, λ) = ln(x) + y + λ(I - Px * x - Py * y).

Step 4: Find the partial derivatives of the Lagrangian function with respect to x, y, and λ, and set them equal to zero.
∂L/∂x = 1/x - λ * Px = 0
∂L/∂y = 1 - λ * Py = 0
∂L/∂λ = I - Px * x - Py * y = 0

Step 5: Solve the system of equations to find the optimal values of x and y.
From ∂L/∂x: λ * Px = 1/x
From ∂L/∂y: λ * Py = 1
Divide the first equation by the second equation to eliminate λ:
(Px * x) / (Py * y) = 1

Step 6: Rearrange the equation to find x* in terms of Px, Py, and I.
x* = Py / Px

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To calculate the probability of winning this game, we need to first figure out all the possible outcomes of flipping three coins at the same time.

There are eight possible outcomes: HHH, HHT, HTH, THH, TTH, THT, HTT, and TTT (where H represents heads and T represents tails). Out of these eight outcomes, only one outcome results in your friend winning (HHH) and the remaining seven outcomes result in you winning. Therefore, the probability of you winning the game is 7/8 or 0.875, which means that you have an 87.5% chance of winning. Keep in mind that this is assuming that the coins are fair and unbiased, meaning that there is an equal chance of getting heads or tails on each flip.

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