. What is the smallest number of seats in a large auditorium that must be occupied in order to be certain that at least three people in attendance have the same first and last initials

Thus, the length of a side of the regular hexagon is 8 centimeters using the Pythagorean theorem.

The key to solving this problem is to realize that the perpendiculars drawn from the interior point to the sides of the hexagon form a right triangle with one leg being the perpendicular and the other leg being a side of the hexagon. We also know that the hexagon is regular, meaning all sides have the same length.

Let's label the length of the side of the hexagon as "x". We can use the Pythagorean theorem to find the length of each perpendicular as follows:

- For the perpendicular that is 4 cm long, we have x^2 = 4^2 + (x/2)^2
- For the perpendicular that is 5 cm long, we have x^2 = 5^2 + (x/2)^2
- For the perpendicular that is 6 cm long, we have x^2 = 6^2 + (x/2)^2
- For the perpendicular that is 8 cm long, we have x^2 = 8^2 + (x/2)^2
- For the perpendicular that is 9 cm long, we have x^2 = 9^2 + (x/2)^2
- For the perpendicular that is 10 cm long, we have x^2 = 10^2 + (x/2)^2

Simplifying each equation and using a bit of algebra, we get:

- 3x^2 = 16^2
- 7x^2 = 25^2
- 12x^2 = 36^2
- 24x^2 = 64^2
- 33x^2 = 81^2
- 40x^2 = 100^2

Solving for x in each equation, we find that x = 8 cm. Therefore, the length of a side of the regular hexagon is 8 centimeters.

In summary, we used the fact that the perpendiculars from an interior point to the sides of a regular hexagon form right triangles to set up equations using the Pythagorean theorem. Solving for the length of a side of the hexagon in each equation, we found that it is 8 cm long.

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The probability that Tamika's result is greater than Carlos' result is $\boxed{\frac{4}{9}}$.

To solve this problem, we can start by finding all the possible sums that Tamika can get by adding two different numbers from the set $\{8,9,10\}$:

- $8+9=17$
- $8+10=18$
- $9+10=19$

Similarly, we can find all the possible products that Carlos can get by multiplying two different numbers from the set $\{3,5,6\}$:

- $3\times5=15$
- $3\times6=18$
- $5\times6=30$

Now we need to compare each sum with each product to see which ones satisfy the condition that Tamika's result is greater than Carlos' result. We can organize this information in a table:

| Tamika's sum | Carlos' product | Tamika's sum > Carlos' product? |
| ------------ | -------------- | ----------------------------- |
| 17           | 15             | Yes                           |
| 17           | 18             | No                            |
| 17           | 30             | No                            |
| 18           | 15             | Yes                           |
| 18           | 18             | No                            |
| 18           | 30             | No                            |
| 19           | 15             | Yes                           |
| 19           | 18             | Yes                           |
| 19           | 30             | No                            |

Out of the 9 possible combinations, there are 4 that satisfy the condition, namely when Tamika gets a sum of 17, 18 (twice), or 19 and Carlos gets a product of 15 or 18. Therefore, the probability that Tamika's result is greater than Carlos' result is $\boxed{\frac{4}{9}}$.

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Answer:  How do you know how many solutions an equation has?

If solving an equation yields a statement that is true for a single value for the variable, like x = 3, then the equation has one soluti

Step-by-step explanation:

The probability that a randomly selected dinner bill will be at least $39.10 is approximately 0.0322.

a. The random variable in this case is the amount of money spent on dinner bills in the local restaurant on a daily basis.

b. To find the probability that a randomly selected bill will be at least $39.10 To do this, we can use the formula  z = (x - μ) / σ

Where

Substituting the values, we get:

z = (39.10 - 28) / 6

z = 1.85

We need to find the probability of getting a z-score of 1.85

The probability can be determined by using a conventional normal distribution table and is as follows:

P(z > 1.85) = 1 - P(z < 1.85) = 1 - 0.9678 = 0.0322

Therefore, the probability that a randomly selected dinner bill will be at least $39.10 is approximately 0.0322.

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The calculated values of the first four terms of the sequence are -4, -2, 2 and 10

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

a1 = -4

an = 2a(n - 1) + 6

Using the above as a guide, we have the following equations

a(2) = 2a1 + 6

a3 = 2a2 + 6

a4 = 2a3 + 6

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

a2 = 2 * -4 + 6 = -2

a3 = 2 * -2 + 6 = 2

a4 = 2 * 2 + 6 = 10

Hence, the first four terms of the sequence are -4, -2, 2 and 10

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Answer:

4c² - 17c + 32

Step-by-step explanation:

To expand (c -2)², use the identity (a - b)² = a² - 2ab + b²

(c - 2)² = c² - 2*c*2 + 2²

            = c² - 4c + 4

Use FOIL method to find (3c + 1)(c -4)

(3c  + 1)(c - 4) = 3c*c  - 3c *4 + 1*c - 1*4

                     = 3c² - 12c + 1c - 4

                      = 3c² - 11c  - 4  {Combine like terms}

7(c - 2)² - (3c + 1)(c -4) = 7*(c²- 4c + 4) - (3c² - 11c - 4)

Multiply each term of c² - 4c + 4 by 7 and each term of 3c² - 11c - 4 by (-1)

                                    = 7c² - 7* 4c + 7*4  - 3c² + 11c + 4

                                     = 7c² - 28c + 28 - 3c² + 11c + 4

                                     = 7c² - 3c² - 28c + 11c + 28 + 4

 Combine like terms,

                                     = 4c² - 17c + 32

Mo's conjecture that a pentagon cannot contain 4 right angles is correct.

A pentagon is a five-sided polygon. In order for a polygon to contain a right angle, it must have at least one interior angle measuring 90 degrees.

The sum of the interior angles of a pentagon is given by the formula (n-2) x 180 degrees, where n is the number of sides. For a pentagon, this formula gives us (5-2) x 180 = 540 degrees.

In order for a pentagon to contain four right angles, the sum of the interior angles that are right angles would have to be 4 x 90 = 360 degrees. However, this is impossible because the remaining interior angles would have to add up to 540 - 360 = 180 degrees.

Since a pentagon only has five interior angles, it is not possible for three of them to add up to 180 degrees, which means it is impossible for a pentagon to contain four right angles.

Therefore, Mo's conjecture that a pentagon cannot contain 4 right angles is correct.

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Answer:

A

Step-by-step explanation:

the answer is A. because the direction of arow is left. and at point 8, the point is hollow.

The vertex of the  parabola is (1, -1), x = 1 as the equation of the axis of symmetry and the roots of the parabola are x = 0 and x = -2

The graph represents the given parabola

As a general rule:

The vertex of a parabola is the minimum or the maximum of a quadratic funtion

In this case, we have

Minimum = (1, -1)

This means that the vertex of the given parabola is (1, -1)

The equation of the axis of symmetry is the x-coordinate of the vertex

In this case, we have

x = 1 as the equation of the axis of symmetry

These are the points where the graph crosse the x-axis

In this case, the roots of the parabola are x = 0 and x = -2

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The half-life of the substance is approximately 17.78 days.

The exponential decay model for the mass of the substance can be written as:

[tex]m(t) = m0 \times e^{(-rt)},[/tex]

where m0 is the initial mass, r is the decay rate parameter (as a decimal), and t is time in days.

If we want to find the half-life of the substance, we need to find the value of t when the mass has decreased to half of its original value (m0/2). In other words, we need to solve the equation:

m(t) = m0/2

[tex]m0 \times e^{(-rt)} = m0/2[/tex]

[tex]e^{(-rt) }= 1/2[/tex]

Taking the natural logarithm of both sides, we get:

-ln(2) = -rt

t = (-ln(2)) / r

Substituting the value of r (0.039), we get:

t = (-ln(2)) / 0.039

t ≈ 17.78 days

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Answer: (x+7)(x+7), 7 plus 7 is 14 and 7 times 7 is 49

The mean, of the sampling distribution of the mean, for samples of size 48 equals 65.

The central limit theorem states that the sampling distribution of the mean of a large number of random samples taken from a population will be approximately normally distributed, regardless of the shape of the population distribution. The mean of the sampling distribution of the mean is equal to the population mean, which in this case is 65.

The standard deviation of the sampling distribution of the mean is equal to the population standard deviation divided by the square root of the sample size. Therefore, the standard deviation of the sampling distribution of the mean for samples of size 48 is 10.2 / √48 ≈ 1.47.

Since the sampling distribution of the mean is approximately normally distributed with a mean of 65 and a standard deviation of 1.47, the mean of the sampling distribution of the mean for samples of size 48 is 65.

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The probability that Max will get the job is 3/10 or 0.3.

To see why, we can use the fact that the sum of the probabilities of all possible outcomes is equal to 1. Let A, B, and C represent the events that the first, second, and third companies respectively get the job.

Then the probability that Max gets the job is equal to the probability of event AB~C (i.e., none of the other companies gets the job).

The probability of A is 1/5, the probability of B is 1/2, and the probability of C is 3/10 (since the sum of the probabilities of all three events is 1). Using the formula for the probability of the intersection of independent events, we have:

P(AB~C) = P(~A) * P(~B) * P(~C) = (4/5) * (1/2) * (7/10) = 14/50 = 0.28

So the probability that Max gets the job is 0.3, or 3/10.

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The probability is 28/45, which simplifies to 0.6222 or approximately 62.22%.

To calculate the probability of both light bulbs being good, we'll use the terms:

sample space, favorable outcomes, and probability.
Sample space:

This is the total number of possible outcomes when selecting two light bulbs.

Since there are 10 light bulbs in the drawer, the sample space is the number of ways to choose 2 light bulbs from 10, which can be calculated using combinations.

So, the sample space is C(10,2) = 10! / (2! * (10-2)!) = 45 possible outcomes.
Favorable outcomes:

These are the outcomes where both light bulbs are good.

There are 8 good light bulbs in the drawer,

so the number of favorable outcomes is the number of ways to choose 2 good light bulbs from the 8, which is C(8,2) = 8! / (2! * (8-2)!) = 28 favorable outcomes.
Probability: To find the probability of both light bulbs being good, divide the number of favorable outcomes by the sample space.

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The perimeter of the triangle is 23 ft.

We have,

We can use the triangle inequality theorem to check whether this triangle can exist.

According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's check whether this condition holds for the sides given:

5 + 7 = 12, which is greater than 11, so the condition holds.

5 + 11 = 16, which is greater than 7, so the condition holds.

7 + 11 = 18, which is greater than 5, so the condition holds.

Therefore, this triangle can exist.

To find the perimeter of the triangle, we simply add up the lengths of the three sides:

Perimeter = 5 + 7 + 11 = 23 ft

Therefore,

The perimeter of the triangle is 23 ft.

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The triangle inequality theorem states that a triangle with sides of 5, 7, and 11 feet is feasible.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

Any two triangle sides' lengths added together must be bigger than the third side's length.

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Thus, the rate at which the boy's shadow is lengthening when he is 20 feet from the light is 3.75 feet/s.

To solve this problem, we need to use similar triangles.

Let's call the length of the boy's shadow "x" and the distance from the streetlight to the boy "y". At the moment he is 20 feet from the light, we have:

y = 20 feet (given)
x + 6 = length of the boy's shadow

We can set up a proportion to relate the length of the boy's shadow to the height of the streetlight:

(x + 6)/x = 15/6

Cross-multiplying and simplifying, we get:

6x + 90 = 15x
9x = 90
x = 10 feet

So at the moment the boy is 20 feet from the light, his shadow is 10 feet long. To find the rate at which his shadow is lengthening, we need to take the derivative of this equation with respect to time:
x + 6 = (y - 15)/y * (y') + 6

where y' is the rate at which the boy is walking away from the light (5 feet/s). Plugging in y = 20 feet and solving for x', we get:
x' = (15/20) * 5 = 3.75 feet/s

Therefore, the rate at which the boy's shadow is lengthening when he is 20 feet from the light is 3.75 feet/s.

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(G) (-9, 12) is the point on the given line.

We can use the point-slope form of the equation of a line to find the equation of the line with slope -1/3 and passing through the point (6, 7):

y - y' = m(x - x'), where m is the slope, and (x', y') is the given point.

Plugging in m = -1/3, x' = 6, and y' = 7, we get:

y - 7 = (-1/3)(x - 6)

Multiplying both sides by -3, we get:

-3y + 21 = x - 6

x + 3y = 27

This is the equation of the line.

To find another point on this line, we can substitute each of the given points into the equation and see which one satisfies it. We can also check the answer choices one by one. Let's start with (F) (-3, 8):

x + 3y = 27

-3 + 3(8) = 21, so this point does not satisfy the equation and is not on the line.

Next, let's try (G) (-9, 12):

x + 3y = 27

-9 + 3(12) = 27, so this point does satisfy the equation and is on the line.

We can stop here and conclude that the answer is (G) (-9, 12). However, just for completeness, let's also check the other answer choices:

(H) (12, 13):

x + 3y = 27

12 + 3(13) = 51, so this point does not satisfy the equation and is not on the line.

(J) (18, -3):

x + 3y = 27

18 + 3(-3) = 9, so this point does not satisfy the equation and is not on the line.

Therefore, the answer is (G) (-9, 12).

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Answer:

the second option

Step-by-step explanation:

because that is supposed to be the median of the data

Answer:

20fish

Step-by-step explanation:

so 1/2 of 16 is 8

or 1/4 of 32 is 8

so 8+32=40

40 1/2 is 20

so 20 fish caught on Monday

Once a model of reality is constructed around certain assumptions, it can be tested to determine its value in predicting outcomes.

Once a model of reality is constructed around certain assumptions, it can be tested to determine its value in predicting outcomes. This involves comparing the model's predictions with actual observed data or outcomes to assess its accuracy and reliability. By testing the model against real-world data, we can evaluate its validity and determine if it accurately represents the underlying reality or phenomenon being studied.

Producing data and graphing equations are related activities that can be part of the process of testing a model, but they are not the primary purpose of the model itself. Producing data involves collecting and generating empirical data that can be used to assess the model's predictions or outcomes. Graphing equations can be a way to visualize the relationships between variables in the model, but it is not the main purpose of the model itself. The primary purpose of constructing a model is to make predictions or generate hypotheses about how a system or phenomenon works, and testing these predictions against real-world outcomes is the key step in evaluating the model's value.

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Since the output of the sigmoid function is 0.64, the predicted probability of the observation being malignant is 0.64. We need to set a threshold probability to classify the observation as malignant or benign.

If we set the threshold probability at 0.5, the observation would be classified as malignant, since the predicted probability (0.64) is greater than the threshold probability. However, the choice of threshold probability depends on the specific problem and the costs associated with false positives and false negatives. So, the predicted category would be malignant if the threshold probability is set at 0.5.

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Let G be a triangulation with n vertices, m edges and f faces. Since G is a triangulation, we know that each face has three edges and each edge is incident to two faces. Therefore, we have:

3f = 2m (1)

Now, let us consider the sum of the degrees of all the vertices in G. Each vertex v is incident to some number of edges, say d(v). Since each edge is incident to exactly two vertices, the sum of the degrees of all the vertices is given by:

2m = Σv∈G d(v) (2)

Using the handshake lemma, we know that the sum of degrees of all vertices in a graph is twice the number of edges, which gives us equation (2).

Now, we can find the average degree of a vertex in G by dividing the sum of degrees of all vertices by the number of vertices:

average degree = (Σv∈G d(v))/n

Substituting equation (2) into the above expression, we get:

average degree = 2m/n

Now, we can use Euler's formula to relate the number of vertices, edges, and faces in a planar graph:

n - m + f = 2

Substituting equation (1) into the above expression, we get:

n - (3f/2) + f = 2

n/2 = f + 2

Substituting the above expression into equation (2), we get:

2m = Σv∈G d(v) <= Σv∈G 6 = 6n

Dividing by n on both sides, we obtain:

2m/n <= 6

Substituting this expression into the expression for the average degree, we have:

average degree = 2m/n <= 6

Therefore, the average degree of a vertex in a triangulation is strictly less than 6.

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On flipping a coin, the number of total possibility for obtaining least one head if we flip the coin seven times are equal to 127.

We have an experiment for flip a coin and keep a record of the results.

Number of total possible outcomes on flipping a coin = 2 = { H, T}

Number of trials or a coin is flipped = 7

We have to determine the number of ways that at least one head if we flip the coin seven times. When we flip a two-sided coin n times, there are [tex]2^ n[/tex] possible outcomes. So, when a coin is flipped 7 times then total possible number of outcomes = 2⁷ = 128

Let's consider an Event A : getting at least one head in seven flips

Complement of this event is getting no head and it is denoted by [tex]A^ c =[/tex]{ T T T T T T T }

and this event will be occur in one way only. So, Number of ways of obtain at least one head if you flip the coin five times = 128 - 1 = 127 ways.

Hence, required value is 127 ways.

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we can expect there to be approximately 696 rats (rounded to the nearest whole) in the bay area of the city after 18 months of growth at a rate of 6% per month.

To solve this problem, we can use the formula for exponential growth:
[tex]P(t) = P(0)e^{rt}[/tex]

where P(0) is the initial population, r is the growth rate (in decimal form), t is the time period, and e is the mathematical constant approximately equal to 2.718.

In this case, P(0) = 310, r = 0.06 (since the population grows at a rate of 6% per month), and t = 18 months. So we have:

[tex]P(18) = 310 e^{(0.06 * 18)}[/tex]
P(18) =696

Therefore, we can expect there to be approximately 696 rats (rounded to the nearest whole) in the bay area of the city after 18 months of growth at a rate of 6% per month.

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The correct answer to the question "The sampling distribution of a statistic probability distribution." is option c) is a probability distribution.

To understand the concept of a sampling distribution, we need to understand what a statistic is. A statistic is a numerical value that describes some aspect of a population. For example, the mean, median, mode, variance, and standard deviation are all examples of statistics.

In statistics, we are often interested in making inferences about a population based on a sample of that population. A sampling distribution is a probability distribution that shows all possible values that a statistic can take on, and the probability of getting each value under the assumption that it had resulted due to chance alone.

The sampling distribution is an important concept in statistics because it allows us to make inferences about a population based on a sample. By knowing the sampling distribution of a statistic, we can calculate the probability of getting a particular value of the statistic under the assumption of random sampling. This information can be used to make decisions about the population, such as whether a hypothesis is supported or rejected.

In conclusion, the sampling distribution of a statistic is a option c) It is a probability distribution that gives the probability of getting each value of a statistic under the assumption it had resulted due to chance alone. It is a critical tool for making inferences about populations based on samples.

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In the marketing class of 50 students who evaluated their instructor using the given scale, the descriptive summary of the survey results indicated that the majority of students rated the instructor as either average or poor, with 45% in each category.

This suggests that the instructor's performance might not have been highly effective or satisfactory for most of the students. Meanwhile, a small percentage of students found the instructor to be good (8%) and even fewer rated them as superior (2%). No students rated the instructor as inferior.

Based on these findings, the conclusion can be drawn that the instructor's performance was perceived as predominantly average or poor by the class, indicating potential areas for improvement in their teaching approach or methods to better cater to students' needs and expectations.

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The horizontal distance along the grid line from the highest grid elevation point to the 97 contour is 52.38 feet.

To solve this problem, we first need to determine the location of the highest grid elevation point on the grid line labeled 96.33 and 97.48.

Let's assume that the highest grid elevation point is located at a distance of x feet from the grid line labeled 96.33. Therefore, the distance from the same point to the grid line labeled 97.48 would be 50 - x feet (as the total length of the grid line is 50 feet).

Now, we need to determine the location of the 97 contour on the same grid line. Let's assume that the 97 contour intersects the grid line at a distance of y feet from the grid line labeled 96.33.

Since the highest grid elevation point is on the same grid line, it must also be on the 97 contour. Therefore, we can set the elevation at the highest point equal to 97 and use this information to solve for x and y.

We can set up two equations based on the information we have:

x² + y² = d² (Equation 1)

x + (50 - x) = y (Equation 2)

where d is the horizontal distance we are trying to find.

We can simplify Equation 2 to:

50 = y

Substituting this into Equation 1, we get:

x² + 50² = d²

Rearranging this equation, we get:

d² = x² + 2500

Now we can substitute 97 for the elevation at the highest point, and solve for x:

(97 - 96.33)/0.01 = x/50

x = 33.5 feet

Substituting this value of x into the equation for d², we get:

d² = (33.5)² + 2500 = 2742.25

Taking the square root of both sides, we get:

d = 52.38 feet (approx.)

Therefore, the horizontal distance along the grid line from the highest grid elevation point to the 97 contour is approximately 52.38 feet.

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The critical points of the function f(x) = 4sin(x)cos(x) in the interval (0, 2π) are:
x = π/4, 3π/4, 5π/4, 7π/4

To find the critical points of the function f(x) = 4sin(x)cos(x) in the interval (0, 2π), we need to find the values of x where the first derivative, f'(x), is equal to 0 or undefined.

First, let's find the derivative using the product rule:

f'(x) = (4sin(x))'(cos(x)) + (4sin(x))(cos(x))'
f'(x) = (4cos(x))(cos(x)) - (4sin(x))(sin(x))
f'(x) = 4(cos^2(x) - sin^2(x))

Now, we set f'(x) = 0 and solve for x:

4(cos^2(x) - sin^2(x)) = 0
cos^2(x) = sin^2(x)

Recall that sin^2(x) + cos^2(x) = 1. So, cos^2(x) = 1 - sin^2(x). Substitute this into the\:

1 - sin^2(x) = sin^2(x)
2sin^2(x) = 1
sin^2(x) = 1/2
sin(x) = ±√(1/2)

So, x = arcsin(±√(1/2)).

We need to find the solutions for x in the interval (0, 2π):

x = π/4, 3π/4, 5π/4, 7π/4

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Answer: hi i am not sure what questions you need but look below.

1. D. The temperature dropped 4 degrees each hour for 5 consecutive days.

2. Brenda simplified the expression correctly.

Step-by-step explanation:

1. the equation is -4(5)=-20.

- means that the number is decreasing or below zero. so If the temp. dropped 4 degrees each hour for 5 days then the expression for that would be -4(5)=-20

2. Brenda is correct because -5x+(2+x)= -5+x+2

You must break the expression up. -5x +x and -5x + 2. Sine -5x+x= -4x and you cant simplify -5x+2 because they dont have the same variable, your answer is -4x+2.

You are welcome

The probability that exactly 20 random numbers will fall in the interval 0.1 to 0.35 is approximately 0.0223, or 2.23%.

To solve this problem, we need to use the binomial probability formula:
[tex]P(X = k) = (n choose k)  p^k  ( (1 - p)^{n-k}[/tex]

where:
- X is the random variable representing the number of successes (random numbers in the interval 0.1 to 0.35)
- k is the number of successes we want (exactly 20)
- n is the total number of trials (100)
- p is the probability of success (the probability that a randomly generated number falls in the interval 0.1 to 0.35)

To find p, we need to determine the fraction of the interval 0 to 1 that is between 0.1 and 0.35:
[tex]p = (0.35 - 0.1) / 1 = 0.25\\p = \frac{0.35-0.1}{1} = 0.25[/tex]

Now we can plug in the values and calculate the probability:

[tex]P(X = 20) = (100 choose 20) (0.25)^{20}   (1-0.25)^{100-20}[/tex]
= 0.0223

Therefore, the probability that exactly 20 random numbers will fall in the interval 0.1 to 0.35 is approximately 0.0223, or 2.23%.

To know more about "Probability" refer here:

https://brainly.com/question/30034780#

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