42 Développer chaque produit, puis réduire les expres-
sions obtenues.
A= 5x-3(x+12)
C = 2x² + x(4x - 5)
B=3x-6+7(2x+4)
D=4x²-x+x(5x-9)

It is true that Jayden's solution is incorrect. It is false that Jayden added inside the parentheses before dividing.

It is false that Jayden substituted the wrong value for a. It is true that Jayden divided 14 by 2 and then added 1. 5. Jayden added inside the parentheses before multiplying. ​

1) The correct solution is

Given,

a ÷ (2 + 1. 5)

Substituting the value of a which is 14

= 14 ÷ (2 + 1. 5)

= 14 ÷ 3.5

= 4

2) As there is no term which needs to be divided so, the second statement is false.

3) Jayden didn't substitute the wrong value of a he just solved the given expression without considering the bracket and divided the 14 which is the value of a by 2.

4) Jyaden divided 14 by 2 and then added 1. 5. Jayden added inside the parentheses before multiplying. ​

i.e. a ÷ (2 + 1. 5)

14 ÷ 2 + 1. 5

7+1.5

8.5

This is the way Jayden solved the equation due to which he arrived at the wrong solution.

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The Correct question is as below

Jayden evaluated the expression a ÷ (2 + 1.5) for a = 14. He said that the answer was 8.5. Choose True or False for each statement.

1. Jayden's solution is incorrect.

2. Jayden added in the parentheses before dividing.

3. Jayden substituted the wrong value for a.

4. Jayden divided 14 by 2 and added 1.5

Given:

81x = 243x

= 243 / 81x

= 3

x = 3

A zero balance in Armando's account can be achieved by subtracting $250 from the account balance. This can be done through a combination of withdrawals and/or transfers out of the account.

Subtracting is a mathematical operation that involves taking one number away from another. It is the inverse of addition, and is represented by the minus sign (-). Subtracting is used to determine the difference between two numbers, or to reduce a number by a given amount. In algebraic terms, subtracting is represented by the equation: a - b = c, where a is the starting number, b is the number to be subtracted, and c is the resulting difference. Subtracting can also be used on fractions, decimals, and other forms of numbers.

For example, Armando could make a $250 withdrawal from the account or transfer $250 to another bank account.

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For every consecutive solid that is dilated by k, the volume is multiplied by k³ thus, Elena's reasons can be used to argue the reasoning.

A dilatation is a transformation that alters the size but not the shape of an item. Every point on an item moves away from or towards a fixed position known as the centre of dilatation when the object is dilated. The scale factor is multiplied by the distance between each location and the centre of dilatation.

Thinking about a cube with s sides. We are aware that this cube's volume is s³. This cube will have a new side length of ks and a new volume of (ks)³ = k³s³ if we dilate it by a factor of k. Hence, as Clare said, the volume has been doubled by k³.

Fo any other solid right now. Similar to how a 2D form may be thought of as being composed of several little rectangles, we can conceive of this solid as being composed of many small cubes. Each of these little cubes will be dilated by a factor of k if we dilate the solid by a factor of k.

Hence, for any solid, if it's dilated by a factor of k, the volume is multiplied by k³.

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1. The retail prices are: $44.36, $34.10 and $67.57

2. The sales prices are: $263.20 and $65.59

Retail price refers to the price at which a product or service is sold to the final consumer by a retailer. This price includes the cost of production, transportation, marketing, and other related expenses, as well as a markup to generate profit for the retailer.

1. New Items - Calculate Retail Prices

  Item                    Wholesale        Mark Up       Calculation              Retail

                              Cost ($)            Percent%                                      Price ($)

a. Louisville             $22.75             95%          22.75+(95%×22.75)  =$44.36

Slugger bat and

batting glove set

b. Wilson                 $11.00                210%         11+(210%×11)              =$34.10

Basketball

c. Element               $26.50              155%      26.50+(155%×26.50) =$67.57

Skateboard

2. Items on sale - Calculate Sale Prices

    Item                 Retail             Discount           Calculation            Sales

                           price ($)          Percent %                                       Price ($)

a. Head              $329.00              20%          329-(20%×329)         $263.20

Snowboard

b. Kelly                $79.99                18%           79.99-(18%×79.99)     $65.59

Ski poles

All data is filled in below table.

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The distribution is skewed to the right.

If the median is to the left of the center of the box and the right whisker is substantially longer than the left whisker in a box plot, then the distribution is skewed to the right.

This means that the majority of the data is clustered on the left side of the box plot and there are some extreme values on the right side that are causing the right whisker to be longer.

The median being to the left of the center of the box indicates that the data is not symmetric and is pulled to the left by the majority of the values.

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An expression which can be used to find the surface area of the triangular prism include the following: C. 2(1/2 · 3 · 4) + 4 · 2 + 3 · 2 + 5 · 2.

In Mathematics, the surface area of a triangular prism can be calculated by using this mathematical expression:

Total surface area of triangular prism = (Perimeter of the base × Length of the prism) + (2 × Base area)

Total surface area of triangular prism = (S₁ + S₂ + S₃)L + bh

where:

By substituting the given parameters into the surface area of triangular prism formula, we have:

Surface area of triangular prism = Area (GFEH + BCEF + BIJC + ABF + ECD)      

Surface area of triangular prism = 4 · 2 + 3 · 2 + 5 · 2 + (¹/₂ · 3 · 4) + (¹/₂ · 3 · 4)  

Surface area of triangular prism = 2(1/2 · 3 · 4) + 4 · 2 + 3 · 2 + 5 · 2.

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Cracking the code: 8 2One number is correct and well placed One number is correct but in the wrong placeTwo numbers are correct but in the wrong place38Nothing is correctCODE8One number is correct but in the wrong placeCracking the code of this sequence of numbers can be a bit tricky, but let's do it step by step. We are given the following clues about the sequence of numbers:

One number is correct and well placed: Since the sequence of numbers is 8 2, we know that the number 8 is in the first position. So the code is either 8 _ _ _ or _ _ _ 8.One number is correct but in the wrong place: This clue tells us that the number 2 is not in the second position of the code, but it is somewhere else.

Therefore, we know that the code is not 8 2 _ _ or _ _ 2 8.Two numbers are correct but in the wrong places: This clue tells us that the code contains the numbers 3 and 8, but they are in the wrong position. Since the code cannot be 8 2 _ _ or _ _ 2 8, we know that the two correct numbers are not in the last two positions. Therefore, the code must be _ 3 8 _ or _ 8 3 _.Nothing is correct: This clue tells us that the code cannot be 3 8 _, 8 3 _, or _ 3 8 because they all contain at least one correct number. Therefore, the code must be _ _ 3 8 or 3 8 _ _.One number is correct but in the wrong place: This clue tells us that the code cannot be 3 8 _, so it must be 8 3 _. Therefore, the code is 8 3 _ _.I hope this helps you crack the code!

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tan nx + tanx1 - tan nxtanx =tan(n+1)x identity holds true in two cases.

To prove the identity:

tan(nx) + tan(x1) - tan(nx)tan(x) = tan((n+1)x)

We'll start with the left-hand side:

tan(nx) + tan(x1) - tan(nx)tan(x)

We can use the identity for the sum of tangents:

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

If we let A = nx and B = x1, then we can write:

tan(nx + x1) = (tan(nx) + tan(x1)) / (1 - tan(nx)tan(x1))

Simplifying the denominator:

tan(nx + x1) = (tan(nx) + tan(x1)) / (tan(nx + x1) - tan(nx)x1)

Multiplying both sides by (tan(nx + x1) - tan(nx)x1):

tan(nx + x1)(tan(nx + x1) - tan(nx)x1) = tan(nx) + tan(x1)

Expanding the left-hand side:

tan²(nx + x1) - tan(nx)x1 tan(nx + x1) = tan(nx) + tan(x1)

Moving all terms to one side:

tan²(nx + x1) - tan(nx + x1)tan(nx)x1 - tan(nx) - tan(x1) = 0

Factoring the quadratic:

(tan(nx + x1) - tan(nx)x1) (tan(nx + x1) - tan(x1)) = 0

So either:

tan(nx + x1) = tan(nx)x1

Or:

tan(nx + x1) = tan(x1)

If we consider the case where tan(nx + x1) = tan(nx)x1, then we can substitute this expression into the left-hand side of the original identity:

tan(nx) + tan(x1) - tan(nx)tan(x)

= tan(nx) + tan(x1) - (tan(nx + x1) / x1)

= tan(nx) + tan(x1) - (tan(nx)x1 / x1)

= tan(nx) + tan(x1) - tan(nx)

= tan(x1)

And this is equal to the right-hand side of the original identity, which is:

tan((n+1)x)

So the identity holds.

If we consider the case where tan(nx + x1) = tan(x1), then we can similarly substitute this expression into the left-hand side of the original identity:

tan(nx) + tan(x1) - tan(nx)tan(x)

= tan(nx) + tan(x1) - (tan(nx + x1) / x1)

= tan(nx) + tan(x1) - (tan(x1) / x1)

= (tan(nx) x1 + tan(x1) - tan(nx)tan(x)) / x1

= tan((n+1)x)

And this is equal to the right-hand side of the original identity, which is:

tan((n+1)x)

So the identity holds in this case as well.

Therefore, we have shown that the identity holds in both cases, and hence it holds in general.

What is Trigonometric identities?

Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.

There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.

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The distance of the line is √17 and the midpoint of the line is ( 5/2 , 2).

Why do you use the word distance?

Distance is the total movement of an object without taking direction into account. Distance can be described as the amount of ground that an object has covered, regardless of its beginning or ending points.

points A (3,-1) B (2,5)

D = √(x₂ - x₁)² + (y₂ - y₁ )²

    = √(2 - 3)² + ( 5- ( -1))²

     = √(-1)² + (4)²

      = √1 + 16

        = √17

  midpoint =   x = 2 + 3/2

                          = 5/2

                   y = 5 - 1/2

                   y = 2

  (x,y) = ( 5/2 , 2)

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The value of m that makes u and v parallel vectors is 5.

if u and v are parallel, we need to find a scalar k such that:

u = kv

k(3i) = I (the i-component of u is equal to k times the i-component of v)

k(-6j) = -aj (the j-component of u is equal to k times the j-component of v, but with a negative sign since the j-component of u is negative)

k(mk) = 5k (the k-component of u is equal to k times the k-component of v)

Simplifying each equation, we get:

[tex]3k = 1\\-6k = -a\\m*k = 5k[/tex]

From the first equation, we get k = [tex]\frac{1}{3}[/tex]. Substituting this value into the second equation, we get a = 2. Finally, substituting k = [tex]\frac{1}{3}[/tex] into the third equation, we get:

m*([tex]\frac{1}{3}[/tex]) = 5*[tex]\frac{1}{3}[/tex])

Simplifying, we get:

m = 5

Parallel vectors are two or more vectors that have the same or opposite direction, regardless of their magnitudes. In other words, parallel vectors are vectors that are either collinear or antiparallel. To determine if two vectors are parallel, one can check if the cross product of the vectors is zero. If the cross product is zero, then the vectors are parallel. If the cross product is nonzero, then the vectors are not parallel.

Parallel vectors are commonly used in various fields such as physics, engineering, and computer science. For example, in physics, parallel vectors are used to describe the motion of objects along a straight line, while in computer graphics, they are used to represent the direction of light and the orientation of objects in a scene.

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Answer: calculate the amount of his refund.

Step-by-step explanation:

An algebraic proof of the triangle inequality is ||u + v|| ≤ ||u|| + ||v||.

Let u and v be vectors in a vector space.

Using the properties of the inner product,

we have[tex],|u + v|² = (u + v) · (u + v)= u · u + u · v + v · u + v · v= |u|² + 2u · v + |v|²[/tex]

Now, take the square root of both sides of the above equation to get

|u + v| = sqrt(|u|² + 2u · v + |v|²) ≤ sqrt(|u|² + 2|u||v| + |v|²)= sqrt(|u|²) + sqrt(2|u||v|) + sqrt(|v|²) = |u| + |v|

Therefore, ||u + v|| ≤ ||u|| + ||v||. The equality holds if and only if u and v are linearly dependent.

Hence, the inequality is sometimes called the strict triangle inequality.

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The measure of angle G is 34° because the sum of the measures of the angles GFE and GF is 180° - 28° = 152° and the measure of angle GFE is 28°, so the measure of angle GF is 152° - 28° = 34°.

An angle is a shape formed when two lines or rays intersect. It is measured in degrees and is used in mathematics and geometry to describe the size and direction of a corner or turn. Angles measure the amount of turn between two lines, and can be described as either acute, obtuse, right, or reflex. Angles are also used to describe the orientation of objects in space, such as the position of the sun or the direction of a car relative to its surroundings.

The measure of angle GFE is 180° minus angle FEG, which is 180° - 28° = 152°. Since the sum of the measures of the angles of a triangle must equal 180°, the measure of angle GF is 180° - 152° = 28°. Therefore, the measure of angle G is 34° because the sum of the measures of the angles GFE and GF is 180° - 28° = 152° and the measure of angle GFE is 28°, so the measure of angle GF is 152° - 28° = 34°.

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The ∠E is 28°. Then what is the angle that is on the opposite side of ∠E It is ∠G.∠G would be 28°.

An angle is a shape formed when two lines or rays intersect. It is measured in degrees and is used in mathematics and geometry to describe the size and direction of a corner or turn. Angles measure the amount of turn between two lines, and can be described as either acute, obtuse, right, or reflex. Angles are also used to describe the orientation of objects in space, such as the position of the sun or the direction of a car relative to its surroundings.

198: line EF and line FG have same length of 6.

A triangle that has to be two sides of the equal length is the  called isosceles triangle.

So your triangle is an isosceles triangle!

An isosceles with the  triangle has two angles on facing each other are the same.

For example, the ∠E is 28°. Then what is the angle that is on the opposite to side of ∠E? It is ∠G.

∠G would be 28°.

So your answer is C. 28°.

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The 12 months of age) as percentiles of the CDC growth chart reference population.

The most likely description of Sarah's weights (at 3, 6, 9, and 12 months of age) as percentiles of the CDC growth chart reference population is: 85th percentile at 3 months; 85th percentile at 6 months; 90th percentile at 9 months; 95th percentile at 12 months.What is percentile in statistics?In statistics, a percentile is a value below which a specific percentage of observations in a group falls. It is used to split up data into segments that represent an equal proportion of the entire group, resulting in a data set split into 100 equal portions, with each portion representing one percentage point. Sarah's weight is in the 85th percentile at 3 months, 85th percentile at 6 months, 90th percentile at 9 months, and 95th percentile at 12 months is a most likely description of her weights (at 3, 6, 9, and 12 months of age) as percentiles of the CDC growth chart reference population.

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Answer:

$735.264

Step-by-step explanation:

First, we need to find 8% of the total payment of the TV:

8% of $799.20

[tex]\frac{8}{100}[/tex] x $799.20

= $63.936

Finally, subtract $63.936 from $799.20:

= $735.264.

Therefore, if I bought a TV and the tax rate is 8% and the total payment of the TV is $799.20, the before tax of the TV is $735.264.

We cannot construct an interval using the normal approximation of survival rate between control and treatment groups because the samples must be random, independent, and their sample sizes must be sufficiently large.

The normal approximation is valid when the sample sizes are large enough to ensure that the sampling distribution of the mean of the variable is approximately normal.

The central limit theorem applies to the distribution of the sample mean when the sample size is large enough, according to the normal approximation.

As a result, the mean difference between the two groups must have a normal distribution. The normal distribution may not be an accurate representation of the underlying distribution of the difference between the two population means in the absence of this requirement, causing the confidence interval to be inaccurate. It will lead to incorrect inferences about the difference in the survival rates of the two groups.

The confidence interval constructed despite this problem will lead to incorrect inferences about the difference in the survival rates of the two groups. This would make it difficult to draw any conclusions based on the findings of this experiment.

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The system of inequalities to represent the constraints of the situation are x ≤ 260, y ≤ 320, and x + y ≤ 360.

A group of two or more linear inequalities are grouped together and graphed on a coordinate plane to discover the solution that concurrently solves all of the inequalities. Each inequality forms a half-plane on the coordinate plane, and the location where all the half-planes overlap is where the system is solved. Each point inside the feasible area meets all of the system's inequalities. This region is known as the feasible region. To identify the optimum solution given a set of constraints, systems of linear inequalities are frequently utilised in optimization issues.

Let us suppose the number of boards of Mahagony sold = x.

Let us suppose the number of black walnut boards sold = y.

According to the given problem the equation can be set as follows:

x ≤ 260 (the company has 260 boards of mahogany available)

y ≤ 320 (the company has 320 boards of black walnut available)

x + y ≤ 360 (the company expects to sell at most 360 boards of wood)

Hence, the system of inequalities to represent the constraints of the situation are x ≤ 260, y ≤ 320, and x + y ≤ 360.

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When a 666-sided fair die is rolled, then the probability of rolling greater than 4 is  0.9940 or 99.40%.

Given that the die is fair and has 666 sides. So, each face of the die will have a probability of 1/666, i.e.,

p(1) = p(2) = ... = p(666) = 1/666.

The probability of rolling greater than 4 is P(roll greater than 4), which is the sum of the probabilities of rolling a 5, 6, 7, 8, 9, ..., 666. So,

P(roll greater than 4) = p(5) + p(6) + p(7) + ... + p(666)

P(roll greater than 4) = (1/666) + (1/666) + (1/666) + ... + (1/666)

(There are 661 terms)P(roll greater than 4) = 661(1/666)

P(roll greater than 4) = 0.9940 (rounded to four decimal places)

Hence, the probability of rolling greater than 4 is 0.9940 or 99.40%.

Alternatively, the probability of rolling greater than 4 is

   1 - P(roll less than or equal to 4)P(roll greater than 4)

= 1 - P(roll less than or equal to 4)P(roll greater than 4)

= 1 - (p(1) + p(2) + p(3) + p(4))P(roll greater than 4)

= 1 - (4/666)P(roll greater than 4)

= 1 - 0.0060P(roll greater than 4)

 = 0.9940 (rounded to four decimal places)

Hence, the probability of rolling greater than 4 is 0.9940 or 99.40%.

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The expression for the number of non-adult sizes is s - 19.

A value or amount is represented by an expression, which is a collection of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. Calculations, complicated mathematical equations, and issues in a variety of disciplines, including science, engineering, economics, and statistics, may all be solved using expressions. Functions that depict a connection between variables, such as sin(x) and log(x), can also be included in expressions. Expressions are frequently employed to simulate real-world circumstances and provide predictions based on mathematical analysis.

Given that the total number od sweatshirts = s.

The number of non-adult sweatshirts can be calculated by:

Number of non-adult sizes = Total number of sweatshirts sold - Number of adult sizes

= s - 19

Hence, the expression for the number of non-adult sizes is s - 19.

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Step-by-step explanation:volume = 0.5 * b * h * length , where b is the length of the base of the triangle, h is the height of the triangle, and length is prism length

The length of the third side of the actual patio is 15.75 feet.

In order to solve issues requiring proportional connections between the sides of two triangles, it is crucial to understand the idea of triangle similarity. If the matching sides and angles of two triangles are identical, then two triangles are said to be similar. When referring to similarities, the symbol is used.

Let us suppose the length of third side = x.

Given that, blueprint of a triangular patio has side lengths 3.5 in., 3.5 in., and 5.25 in.

Thus, the blueprint and the actual patio form two similar triangles.

The sides of the similar triangles are equal in ratio thus,

3.5/5.25 = 10.5/x

3.5x = 5.25 * 10.5

x = (5.25 * 10.5) / 3.5

x = 15.75

Hence, the length of the third side of the actual patio is 15.75 feet.

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Answer:

a = -9

Step-by-step explanation:

2(a+3) = -12

2a + 6 = -12

2a = -18

a = -9

Let's Check

2(-9 + 3) = -12

2(-6) = -12

-12 = -12

So, a = -9 is the correct answer.

Answer:

Let's call the first number x and the second number y.

From the problem, we know that:

To solve for x and y, we can start by expressing one variable in terms of the other, using one of the equations.

Rearranging the first equation to solve for x, we get:

x = 425 - y

Now we can substitute this expression for x into the second equation, and solve for y:

0.2x = 0.3y (substituting x = 425 - y)

0.2(425 - y) = 0.3y (distributing the 0.2)

85 - 0.2y = 0.3y (combining like terms)

85 = 0.5y (adding 0.2y to both sides)

y = 170 (dividing both sides by 0.5)

So the second number is 170. To find the first number, we can substitute this value back into the first equation:

x + y = 425 (substituting y = 170)

x + 170 = 425

x = 255

Therefore, the two numbers are 255 and 170.

Answer:

28 is the ans of 252÷9here we go

252 divided by 9 equals 28.

To divide 252 by 9, you can use long division, which involves dividing the number in steps until there is no remainder left.

Here's the step-by-step process:

Write down the dividend (252) and the divisor (9), and set up the long division format:

9 | 252

Look at the leftmost digit of the dividend (2) and see if it's divisible by the divisor (9). Since 2 is less than 9, we bring down the next digit (5) to the right of 2, making it 25.

9 | 252

2

Divide the new number (25) by the divisor (9). The result is 2, which is the first digit of the quotient. Multiply this result by the divisor (2 x 9 = 18) and write it below the 25, then subtract it from 25:

9 | 252

25

18

--

7

Bring down the next digit (2) from the dividend to the right of the remainder (7), making it 72. Now, divide 72 by 9, which gives you 8. Multiply this result by the divisor (8 x 9 = 72) and write it below the 72, then subtract it from 72:

9 | 252

25

18

--

72

72

---

0

There is no remainder left, and the dividend has been completely divided. The quotient is the result of the division, which is 28.

Therefore, 252 divided by 9 equals 28.

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The complement of the angle is 38 degrees and the supplement of the angle is 128 degrees.

The given angle is

Angle = 52 degrees

To find the complement of an angle, we need to subtract its measure from 90 degrees.

Therefore, the complement of an angle with a measure of 52 degrees is:

90 degrees - 52 degrees = 38 degrees

So, the complement of the angle shown is 38 degrees.

To find the supplement of an angle, we need to subtract its measure from 180 degrees.

Therefore, the supplement of an angle with a measure of 52 degrees is:

180 degrees - 52 degrees = 128 degrees

So, the supplement of the angle shown is 128 degrees.

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The solution for X in equation 30=5(X+5)X is X= 1.

To solve the equation, we can start by distributing the 5 on the right-hand side of the equation, which gives us:

30 = 5X + 25X

Combining like terms, we get:

30 = 30X

Dividing both sides by 30, we get:

X = 1

However, we need to check whether this value satisfies the original equation. Plugging X=1 into the equation gives us:

30 = 5(1+5)(1)

30 = 5(6)

30 = 30

Therefore, the only valid solution is X=1.

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Existence and Uniqueness Theorem the existence and uniqueness theorem is the most critical theorem in differential calculus. The theorem addresses how the existence and uniqueness of a solution to a first-order differential equation are affected by conditions such as continuity or Lipschitz continuity.

Determine if the existence and uniqueness theorem does or does not guarantee the existence and uniqueness of a solution to each of the following initial value problems.

1. The differential equation is

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

       The condition of the theorem is fulfilled: The differential equation is continuous and the partial derivative   [tex]\frac{\partial f}{\partial y}=\frac{-1}{2\sqrt{x-y}}[/tex]  is continuous.

            Therefore, the theorem guarantees the existence and uniqueness of the solution to the initial value problem.

2. The differential equation is

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

         The condition of the theorem is not fulfilled.

            [tex]\frac{\partial f}{\partial y}=\frac{-1}{2\sqrt{x-y}}[/tex] is not defined at [tex]x=y .[/tex]

            Therefore, the theorem does not guarantee the existence and uniqueness of the solution to the initial value problem.

3. The differential equation is  [tex]y\frac{dy}{dx}=x-1[/tex] The condition of the theorem is fulfilled: The differential equation is continuous and the partial derivative [tex]\frac{\partial f}{\partial y}=y[/tex] is continuous.

     Therefore, the theorem guarantees the existence and uniqueness of the solution to the initial value problem.

4. The differential equation is

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

       The condition of the theorem is not fulfilled   [tex]\frac{\partial f}{\partial y}=y[/tex]  and is not defined at  y=0

    Therefore, the theorem does not guarantee the existence and uniqueness of the solution to the initial value problem.

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The probability that the sample contains exactly 4 minorities is 0.0577

We know that 25% of the employees of a large company are minorities. So, from any set of N employees that we ranodmly select from that large company, we can estimate that the number of minorities there is:

0.25*N

Now, we want to find the probability that the sample contains exactly 4 minorities.

Then we need to compute:

[tex]P = 0.25^4*0.75^3[/tex]

We also need to count the permutations for that probability, then we need to add the factor:

[tex]P = 0.25^4*0.75^3*(\frac{7!}{4!*3!} )[/tex]

Solving that we will get:

[tex]P = 0.25^4*0.75^3*(\frac{7!}{4!*3!} )\\\\P = 0.25^4*0.75^3*(\frac{7*6*5}{3*2} )\\\\P = 0.0577[/tex]

That is the probability.

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After all the linking that will happen, there will be only one train left, moving at the speed of the slowest initial train.

If the three equally spaced trains had started in a different order, the final configuration would still be the same - one train moving at the speed of the slowest initial train.

On average, there will always be only one train left, regardless of the initial ordering of the trains. This is because after all the linking, the train with the slowest speed will always be at the front of the train and no other train can pass it.

If there are four equally spaced trains initially, after all the linking, there will be one train left moving at the speed of the slowest initial train. This is because any two faster trains that link up will then be slower than the train behind them, so the only train left will be the slowest one.

Similarly, if there are five or more equally spaced trains initially, after all the linking, there will be only one train left moving at the speed of the slowest initial train.

Therefore, regardless of the number of equally spaced trains initially, there will always be only one train left after all the linking.

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