How many four-letter sequences are possible that use the letters b, r, j, w once each? sequences

The required answer is the probability of getting the sequence BD is 1/30.

a) To find the probability that one chip will be A and one will be E, we need to first determine the total number of possible outcomes. Since we are selecting two chips without replacement, there are 6 ways to choose the first chip and 5 ways to choose the second chip. Therefore, there are 6 x 5 = 30 possible outcomes.

Next, we need to determine the number of outcomes where one chip is A and one chip is E. There are two ways this can happen: A can be the first chip and E can be the second, or E can be the first chip and A can be the second. Therefore, there are 2 possible outcomes where one chip is A and one chip is E.

The probability of getting one chip that is A and one that is E is therefore 2/30, or 1/15.

b) To find the probability that the first chip will be F, we again need to determine the total number of possible outcomes. Since there are 6 chips, there are 6 ways to choose the first chip.

Out of those 6 possible outcomes, only 1 of them results in the first chip being F. Therefore, the probability of the first chip being F is 1/6.

c) To find the probability that the first chip will be B and the second will be D, we again need to determine the total number of possible outcomes. There are 6 ways to choose the first chip and 5 ways to choose the second chip, giving us 6 x 5 = 30 possible outcomes.

Out of those 30 possible outcomes, only 1 of them results in the first chip being B and the second chip being D (BD).

Therefore, the probability of getting the sequence BD is 1/30.

a) To find the probability that one chip will be A and one will be E, you first need to determine the total number of possible outcomes when selecting two chips without replacement. There are 6 choices for the first chip and 5 choices for the second chip, so there are 6 x 5 = 30 possible outcomes.

Now, there are 2 ways to select chips A and E: AE or EA. So the probability of selecting one A and one E is:
P(A and E) = Number of favorable outcomes (AE or EA) / Total possible outcomes = 2/30 = 1/15

b) To find the probability that the first chip will be F, you need to consider that there are 6 chips in total. Only 1 of them is F, so the probability is:
P(First chip is F) = Number of favorable outcomes (F) / Total possible outcomes = 1/6

c) To find the probability that the first chip will be B and the second chip will be D, you need to consider the possible outcomes. There is only 1 favorable outcome: selecting B first and then D. So the probability is:
P(First chip is B and second chip is D) = Number of favorable outcomes (BD) / Total possible outcomes = 1/30

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The Set corresponding to (AUB)nc is Option C. (3, 4, 5, 7, 9).

(AUB)nc represents the complement of the union of sets A and B. To find this set, we first need to find the union of sets A and B, which is the set of all elements that are in either A or B or both.

Set A contains all prime numbers, so A = (2, 3, 5, 7, 11, ...). Set B contains (4, 7, 9, 11, 13, 14). Taking the union of sets A and B gives us:

AUB = (2, 3, 4, 5, 7, 9, 11, 13, 14)

The complement of this set (denoted by nc) contains all elements that are not in this set. Therefore, (AUB)nc contains all elements that are not in the union of sets A and B.

Option A contains 13, which is in AUB. Option B contains 4 and 7, which are also in AUB. Option D contains all elements in AUB. Therefore, the correct answer is option C, which contains (3, 4, 5, 7, 9) and does not contain any elements that are in A or B.

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Consider a homogeneous linear system with a 4 by 9 coefficient matrix. The coefficient matrix has rank at most equal to 4 and at least 5 free variable columns.

This is because the rank of a matrix cannot exceed the number of rows or columns it has. In this case, the matrix has 4 rows, so the rank cannot exceed 4.

Additionally, the number of free variable columns can be found by subtracting the rank of the matrix from the number of columns. In this case, there are 9 columns, so subtracting the maximum rank of 4 gives us 5 free variable columns.

Therefore, The coefficient matrix has rank at most equal to 4 and at least 5 free variable columns.

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(a)  The estimated value of f(61) is 384.

(b) The estimated value of f''(58) is 0.

(a) Using linear approximation, we have:

f(61) ≈ f(60) + f'(60)(61 - 60)

Substituting the given values, we get:

f(61) ≈ 378 + 6(1)

≈ 384

Therefore, the estimated value of f(61) is 384.

(b) Since f is a differentiable function, we can use the second derivative test to estimate f''(58) as follows:

f''(58) ≈ lim h → 0 [tex](f(58 + h) - 2f(58) + f(58 - h)) / h^2[/tex]

Using linear approximation, we have:

f(58 + h) ≈[tex]f(58) + f'(58)h + f''(58)h^2/2[/tex]

f(58 - h) ≈ [tex]f(58) - f'(58)h + f''(58)h^2/2[/tex]

Substituting these values, we get:

f''(58) ≈ lim h → 0[tex][ (f(58) + f'(58)h + f''(58)h^2/2) - 2f(58) + (f(58) - f'(58)h + f''(58)h^2/2) ] / h^2[/tex]

Simplifying and rearranging terms, we get:

f''(58) ≈ lim h → 0[tex][ (f(58 + h) - 2f(58) + f(58 - h)) /[/tex][tex]h^2 - f''(58)h^2][/tex]

Taking the limit as h approaches 0, we get:

f''(58) ≈ f''(58)(0) = 0

Therefore, the estimated value of f''(58) is 0.

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The price of the phone increased by 10% from its initial value c, as indicated by the formula c(1.10) for the previous month's price.

A 10% increase would mean adding 10% of c to c itself if the cost of the phone had been c dollars two months prior. One way to put this is as.

Price last month [tex]= c + 0.10c[/tex]

Simplifying this expression, we can factor out c to get:

Price last month [tex]= c(1 + 0.10)[/tex]

Further, streamlining allows us to assess the expression enclosed in brackets:

If the phone cost c dollars two months ago, then a 10% increase would be 0.1c dollars.

The total of the initial price and the increase, which is:

[tex]c + 0.1c[/tex]

Price last month [tex]= c(1.10)[/tex]

Therefore, The price of the phone increased by 10% from its initial value c, as indicated by the formula [tex]c(1.10)[/tex] for the previous month's price.

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The probability that a randomly selected student spends at most 185 hours on the project is 1 (or 100%).

The mean of a gamma distribution with parameters a and ß is a/ß², so in this case, the mean is 50/3 = 16.67 hours.

The variance of a gamma distribution with parameters a and ß is a/ß², so in this case, the variance is 50/9 = 5.56 hours. Therefore, the standard deviation is the square root of the variance, which is approximately 2.36 hours.

We can convert X to a standard normal variable Z using the formula:

Z = (X - μ) / σ

where μ is the mean of X and σ is the standard deviation of X. Substituting in the values we calculated in Step 1, we get:

Z = (X - 16.67) / 2.36

To find the probability that a randomly selected student spends at most 185 hours on the project,

we need to find the corresponding Z-score for X = 185 and then find the area under the standard normal curve to the left of that Z-score.

Z = (185 - 16.67) / 2.36 = 69.53

Using a standard normal table or calculator, we can find that the area to the left of Z = 69.53 is essentially 1. Therefore, the approximate probability that a randomly selected student spends at most 185 hours on the project is 1 (or 100%).

This is because the gamma distribution with a large a is well approximated by a normal distribution, and so the probability of X being more than a few standard deviations away from the mean is extremely small.

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The confidence interval of 99% for μ₂ - μ₁ for the given mean and standard deviation is equal to (23.7377, 49.7713).

Confidence interval = 99%  

Confidence interval for μ₂ - μ₁, we need to follow these steps,

Calculate the sample mean difference and the standard error of the mean difference.

Sample mean difference

= ¯x₂ - ¯x₁

= 162.75 - 126

= 36.75

Standard error of the mean difference

= √[(s₁^2/n₁) + (s₂^2/n₂)]

= √[(8.062^2/5) + (3.5^2/4)]

= 4.0065 (rounded to four decimal places)

The t-value for a 99% confidence level with degrees of freedom

= n₁ + n₂ - 2

= 5 + 4 - 2

= 7.

Using a t-distribution table attached ,

The t-value for a 99% confidence level with 7 degrees of freedom is 3.250.

Margin of error

= t-value x standard error of the mean difference

= 3.250 x 4.0065

= 13.0213 (rounded to four decimal places)

Confidence interval

= Sample mean difference ± Margin of error

= 36.75 ± 13.0213

= (23.7377, 49.7713)

Therefore, the 99% confidence interval for μ₂ - μ₁ is (23.7377, 49.7713).

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The formula to find the total number of trees, t, in r rows is [tex]t = 10 \times r[/tex]

Sequence and series - A sequence is a list of items/objects which have been arranged in a sequential way. A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

If there are 10 trees planted in each row, then the total number of trees in one row is 10. To find the total number of trees in r rows, we need to multiply the number of trees in one row (10) by the number of rows (r):

[tex]t = 10 \times r[/tex]

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If the trees are planted in rows of 10, then the total number of trees in r rows can be found using the formula: t = 10r

A sequence is an ordered list of numbers or other mathematical objects, such as functions or geometric figures, that follow a certain pattern or rule. A sequence can be finite or infinite and can be specified using a formula or a recursive rule.

A series is a sum of the terms in a sequence, typically written using the sigma notation Σ

where t is the total number of trees and r is the number of rows. This formula simply multiplies the number of trees in each row (which is 10) by the number of rows, to get the total number of trees.

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The given differential equations can be rewritten in matrix-vector form as dX/dt = AX, where X = [x₁, x₂]ᵀ and A = [[2, 1], [1, 2]].

To rewrite the given differential equations in matrix-vector form, follow these steps:

1. Identify the dependent variables, x₁ and x₂, and arrange them into a column vector, X. This gives X = [x₁, x₂]ᵀ.

2. Identify the coefficients of x₁ and x₂ in the given differential equations. For dx₁/dt = 2x₁ + x₂ and dx₂/dt = x₁ + 2x₂, these coefficients are 2, 1, 1, and 2.

3. Arrange the coefficients into a matrix A, with rows corresponding to the order of the dependent variables. This gives A = [[2, 1], [1, 2]].

4. Write the matrix-vector equation dX/dt = AX. This represents the original system of differential equations in matrix-vector form.

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Answer:

[tex]x=\frac{7}{5}[/tex]

Step-by-step explanation:

Using degrees, the formula for arc length is [tex]s= r\theta[/tex], where s is the arc length, r is the radius, and θ is the central angle of the arc in radians.

As we have the length of the arc and we are looking for the central angle, we make θ the unknown and solve for it:

[tex]7=5x[/tex]

We simply divide 5 into both sides to conclude that,

[tex]x=\frac{7}{5}[/tex]

The linear equation in point-slope form is:

y + 6 = (3/4)*(x + 4)

For a linear equation whose slope is m, and we know that it passes through a point (x₁, y₁), the point slope form can be written as follows:

y - y₁ = m*(x - x₁)

Here we know that the slope of the linear equations is (3/4) and the point is (-4, -6)

Then the linear equation in the point slope form can be written as:

y + 6 = (3/4)*(x + 4)

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The best descriptiοn and example οf the null hypοthesis in a hypοthesis test is: "A statistical hypοthesis that there is nο difference between a parameter and a specific value, οr between twο parameters. Example: Hοu - 90"

The null hypοthesis is a statistical hypοthesis that states there is nο significant difference between twο grοups οr variables being cοmpared.

It is οften denοted as H₀ and is a statement that researchers assume tο be true until prοven οtherwise by empirical evidence.

Frοm the given οptiοns

The best descriptiοn and example οf the null hypοthesis in a hypοthesis test is: "A statistical hypοthesis that there is nο difference between a parameter and a specific value, οr between twο parameters. Example: Hοu - 90"

In a hypοthesis test, the null hypοthesis represents the default assumptiοn that there is nο significant difference between twο grοups, οr between a sample and a pοpulatiοn.

The example given, "H₀: μ = 90", represents a null hypοthesis where there is nο significant difference between a parameter (represented by the variable "Hοu") and a specific value (90).

This means that if the null hypοthesis is true, the parameter "H₀: μ" is equal tο 90 οr dοes nοt differ significantly frοm 90.

Hence,

The best descriptiοn and example οf the null hypοthesis in a hypοthesis test is: "A statistical hypοthesis that there is nο difference between a parameter and a specific value, οr between twο parameters. Example: Hοu - 90"

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The matrix of the linear transformation t : R4 → R3 defined by t(x1, x2, x3, x4) = (2x1 3x2 x4, 5x1 9x3 − x4, 4x1 2x2 − x3 7x4) is:

| 2 0 0 0 |

| 0 0 5 -1 |

| 4 8 -2 7 |

To determine the matrix of the linear transformation t : R4 → R3, we need to find the image of the standard basis vectors under the transformation t. The standard basis vectors of R4 are e1 = (1, 0, 0, 0), e2 = (0, 1, 0, 0), e3 = (0, 0, 1, 0), and e4 = (0, 0, 0, 1).

Applying the transformation t to each of these vectors, we get:

t(e1) = (2, 0, 4)

t(e2) = (0, 0, 8)

t(e3) = (0, 5, -2)

t(e4) = (0, -1, 7)

Thus, the matrix of the linear transformation t with respect to the standard bases of R4 and R3 is:

| 2 0 0 0 |

| 0 0 5 -1 |

| 4 8 -2 7 |

Each column of this matrix represents the image of the corresponding basis vector of R4, expressed as a linear combination of the basis vectors of R3.

Note that the matrix has 3 rows and 4 columns, reflecting the fact that the transformation maps R4 to R3.

The first column represents the image of the first basis vector e1, which is (2, 0, 4) in R3.

Similarly, the second, third, and fourth columns represent the images of the basis vectors e2, e3, and e4, respectively.

Therefore, the matrix of the linear transformation t : R4 → R3 defined by t(x1, x2, x3, x4) = (2x1 3x2 x4, 5x1 9x3 − x4, 4x1 2x2 − x3 7x4) is:

| 2 0 0 0 |

| 0 0 5 -1 |

| 4 8 -2 7 |

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The tangential component of acceleration is c. aT = 4t/√4t²+5.

We need to follow these steps:

1. Calculate the first derivative of r(t) to get the velocity vector v(t).
2. Calculate the second derivative of r(t) to get the acceleration vector a(t).
3. Calculate the magnitude of the velocity vector |v(t)|.
4. Calculate the tangential component of acceleration aT by finding the dot product of a(t) and v(t), and then dividing by the magnitude of the velocity vector |v(t)|.

Let's go through these steps:

1. r(t) = ti + t²j + 2tk
  v(t) = dr(t)/dt = (1)i + (2t)j + (2)k

2. a(t) = dv(t)/dt = (0)i + (2)j + (0)k

3. |v(t)| = √(1² + (2t)² + 2²) = √(1 + 4t² + 4) = √(4t² + 5)

4. aT = (a(t) • v(t)) / |v(t)| = ((0)(1) + (2)(2t) + (0)(2)) / √(4t² + 5) = (4t) / √(4t² + 5)

So, the tangential component of acceleration is:

aT = 4t / √(4t² + 5)

This corresponds to option c. aT = 4t/√4t²+5.

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A geometric shape that could be used to model a building with a quadrilateral base and triangular sides is a pyramid.

A geometric shape is a two-dimensional or three-dimensional object that can be described using mathematical formulas and properties. Examples of two-dimensional geometric shapes include squares, circles, triangles, and rectangles. Examples of three-dimensional geometric shapes include cubes, spheres, cylinders, and cones.

Geometric shapes are used in many different fields, including mathematics, science, architecture, engineering, and art. They are important for understanding spatial relationships and for solving problems related to measurement, area, volume, and other geometric properties.

Specifically, the shape would be a triangular pyramid, with the base being a quadrilateral and the sides being triangles.

A cone also has a circular base and curved sides, while a cylinder has circular bases and straight sides. A sphere is a three-dimensional shape with a curved surface, and would not be an appropriate shape to model a building with a quadrilateral base and triangular sides.

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Answer:

The answer is B. Pyramid I just took the test and got it correct.

The length of side VW is equal to 37 units.

In Mathematics and Geometry, the perimeter of a rectangle can be calculated by using this mathematical equation (formula);

P = 2(L + W)

Where:

By substituting the given side lengths into the formula for the perimeter of a rectangle, we have the following;

P = 2(4z + 1 + 3z + 2)

132 = 2(7z + 3)

132 = 14z + 6

z = 126/14

z = 9

VW = 4z + 1 = 4(9) + 1 = 37 units.

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Answer:

2.80

Step-by-step explanation:

35p = £0.35

8 × £0.35 = £2.80

The data set is a Population, because it is a collection of salaries for all baseball players in the league.Therefore option D is correct.

To determine whether the data set is a population or a sample:

Population, because it is a collection of salaries for all baseball players in the league.

The data set includes every single baseball player's salary in the league, which makes it a population.

It is not a sample because it includes every individual in the group being studied, rather than just a subset of them.

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To determine if the integral is convergent or divergent, we need to evaluate the given integral. Integers are a type of number in mathematics that include both positive and negative whole numbers, as well as zero.

Here's the integral you provided:
∫(from 0 to 3) [(30x^2 - 7x)/10] dx

First, let's simplify the integer and by dividing each term by 10:
∫(from 0 to 3) [3x^2 - (7/10)x] dx

Now, we need to find the antiderivative of the simplified integrand:

Antiderivative of 3x^2 is x^3, and the antiderivative of (7/10)x is (7/20)x^2.

So, the antiderivative of the integrand is:
x^3 - (7/20)x^2

Next, we'll evaluate the antiderivative at the limits of integration (0 and 3):
(x^3 - (7/20)x^2) | (from 0 to 3)
= (3^3 - (7/20)(3^2)) - (0^3 - (7/20)(0^2))
= (27 - (7/20)(9)) - (0 - 0)
= 27 - (63/20)

Now, since we got a finite value for the integral, we can conclude that the integral is convergent.

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Answer:

Step-by-step explanation:

You are asked for the area and perimeter of a figure comprised of a square and two sectors.

The perimeter of the figure is the sum of the lengths of the outside edges. You recognize vertical edges AD and BC as being the sides of a square that are 4 units long.

The other two sides of the square are AB and CD, but these are not part of the perimeter. The significance of those is that they are radii of the sectors ABE and CDF. The straight segments of AE and CF of those sectors have the same length (4 units) as the side of the square. Those straight segments are part of the perimeter.

In effect, the four straight segments of the perimeter are all 4 units.

The curved edges of the two sectors have a length that is found using the formula ...

  s = rθ

where r is the sector radius, and θ is the central angle in radians.

The angle is shown as 30°, which is 30°(π/180°) = π/6 radians. The radius is the square side length, 4, so each curved line has length ...

  s = (4)(π/6) = 2/3·π

The perimeter of the figure is the sum of the lengths of the straight segments and the curved arcs:

  P = 4(4 units) +2(2/3π units) = 16 +4/3π units ≈ 20.19 units

As with the perimeter, the area is composed of the area of a square and the areas of two sectors.

The area of the square is the square of its side length:

  A = s²

  A = (4 units)² = 16 units²

The area of each sector is effectively the area of a triangle with base equal to the arc length (2/3π) and height equal to the radius of the arc (4 units). The sector area is ...

  A = 1/2rs

  A = 1/2(4 units)(2/3π units) = 4/3π units²

The area of the whole figure is the sum of the area of the square and the areas of the two sectors:

  A = square area + 2×(sector area)

  A = 16 units² + 2×(4/3π units²) = (16 +8/3π) units² ≈ 24.38 units²

__

Additional comment

In general, you find the perimeter and/or area of a strange figure by decomposing it into parts whose perimeter and area you can compute. (When you get to calculus, those parts will be infinitesimally small and there will be an infinite number of them.) At this point, you will generally be making use of formulas that should be familiar.

The formula for the area of a sector is usually written ...

  A = 1/2r²θ

Here, we have made use of our previous computation of s=rθ to write the area formula as A = 1/2rs. The similarity to the triangle area formula is not accidental.

The Reverse Polish Notation of the expression A * B )/( C * D is: AB* CD* /

To express the given expression A * B )/( C * D in Reverse Polish Notation (RPN):

You would follow these steps:

STEP 1: Identify the operators and operands in the expression: A * B, /, and C * D
STEP 2: Convert the sub-expressions to RPN:
  - A * B becomes AB*
  - C * D becomes CD*
STEP 3: Combine the RPN sub-expressions with the remaining operator, /:
  - AB* CD* /

So, the Reverse Polish Notation of the expression A * B )/( C * D is: AB* CD* /

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We can express the number 0.94 as a ratio of integers by recognizing the repeating pattern in its decimal expansion and converting it to a fraction with a denominator of 100. The resulting fraction is 94/100, which simplifies to 47/50.

To express the number 0.94 as a ratio of integers, we need to find a pattern in its decimal expansion. As we can see, the decimal expansion of 0.94 repeats after the second digit, with the repeating pattern of 94. Therefore, we can write 0.94 as 94/100 or simplified to 47/50.

To understand this concept further, we can think of decimals as a shorthand way of writing fractions. A decimal is just another way to write a fraction with a denominator of 10, 100, 1000, etc. For example, 0.5 is equivalent to 5/10 or simplified to 1/2. In the case of 0.94, we can see that it is equal to 94/100, which can be further simplified to 47/50 by dividing both the numerator and denominator by 2.

The process of converting a decimal to a fraction can be useful in many different areas of math, including algebra, geometry, and calculus. It is important to understand this concept because fractions are an essential part of math and are used in many real-life situations, such as cooking, budgeting, and measurement.

In summary, we can express the number 0.94 as a ratio of integers by recognizing the repeating pattern in its decimal expansion and converting it to a fraction with a denominator of 100. The resulting fraction is 94/100, which simplifies to 47/50.

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Answer:

C

Step-by-step explanation:

Sin is opposite divided by hypotenuse, so that would be 21 divided by 35.

The solution set for the inequality 2 - 4x > 6 is x < -1, expressed in interval notation as (-∞, -1).

To solve the inequality 2 - 4x > 6, we need to isolate the variable x on one side of the inequality.

2 - 4x > 6

Subtract 2 from both sides:

-4x > 4

Divide both sides by -4, remembering to flip the inequality since we are dividing by a negative number:

x < -1

Therefore, the solution set for the inequality 2 - 4x > 6 is x < -1, expressed in interval notation as (-∞, -1).

To graph this solution set, we can draw a number line and shade everything to the left of -1.

<=================|----------->

                 -1    

The shaded part of the number line represents the solution set (-∞, -1).

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The ordered pairs in S are {(4, 4), (5, 5)}, and the ordered pairs in S–1 are {(4, 4), (5, 4), (5, 5), (6, 5)}.

The relation S is defined as x S y ⇔ x | y, which means that x divides y.

Using this definition, we can determine which ordered pairs are in S:

(3, 4) is not in S, since 3 does not divide 4

(3, 5) is not in S, since 3 does not divide 5

(3, 6) is not in S, since 3 does not divide 6

(4, 4) is in S, since 4 divides 4

(4, 5) is not in S, since 4 does not divide 5

(4, 6) is not in S, since 4 does not divide 6

(5, 4) is not in S, since 5 does not divide 4

(5, 5) is in S, since 5 divides 5

(5, 6) is not in S, since 5 does not divide 6

Therefore, the ordered pairs in S are:

{(4, 4), (5, 5)}

The relation S–1 is the inverse of S. An ordered pair (a, b) is in S–1 if and only if (b, a) is in S. In other words, (a, b) is in S–1 if and only if b divides a.

Using this definition, we can determine which ordered pairs are in S–1

(4, 3) is not in S–1, since 4 does not divide 3

(5, 3) is not in S–1, since 5 does not divide 3

(6, 3) is not in S–1, since 6 does not divide 3

(4, 4) is in S–1, since 4 divides 4

(5, 4) is in S–1, since 5 divides 4

(6, 4) is not in S–1, since 6 does not divide 4

(4, 5) is not in S–1, since 4 does not divide 5

(5, 5) is in S–1, since 5 divides 5

(6, 5) is in S–1, since 6 divides 5

Therefore, the ordered pairs in S–1 are  {(4, 4), (5, 4), (5, 5), (6, 5)}

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The given question is incomplete, the complete question is:

Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the “divides” relation. That is, for all (x, y) ∈ A x B,

x S y ⇔ x | y.

State explicitly which ordered pairs are in S and S–1.

The derivative of the given function h(x) = 6x^(-4) - 3x^(-6) is h'(x) = -24x^(-5) + 18x^(-7).

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The sample size increases, but the standard deviation may increase, decrease, or stay the same depending on the probability of success.

False.

The mean (or expected value) of a binomial distribution is given by the formula np, where n is the sample size and p is the probability of success. So as the sample size increases, the mean of the distribution increases proportionally, assuming the probability of success remains constant.

However, the standard deviation of a binomial distribution is given by the formula sqrt(np(1-p)). As the sample size increases, the standard deviation does not necessarily increase. In fact, it can decrease if the probability of success is small or large, and increase if the probability of success is close to 0.5. This is because the variance of the binomial distribution is given by np(1-p), which has a maximum value at p = 0.5. When the probability of success is close to 0 or 1, the variance decreases as the sample size increases because the outcome becomes more predictable. Conversely, when the probability of success is close to 0.5, the variance increases as the sample size increases because there is greater variability in the outcomes.

In summary, the mean of a binomial distribution always increases as the sample size increases, but the standard deviation may increase, decrease, or stay the same depending on the probability of success.

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The left out money = 20- 18.25

= $1.75

Given that:

2 bottles of multivitamin = $3.75 x 2 = $7.5

1 bottle of vitamin D = $4.85

2 Vitamin C bottles = $2.95 x 2 = $5.9

The total = 7.5 + 4.85 +5.9 = $18.25

If $20 had to be spent,

The left out money = 20- 18.25

= $1.75

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