A newsletter publisher believes that under 69% of their readers own a Rolls Royce. Is there sufficient evidence at the 0.10 level to substantiate the publisher's claim

The probability of Suzan getting one of each color other than fluorescent pink, given that she gets the fluorescent pink one, is 108/1001 or approximately 0.108.

To find the probability of Suzan getting one of each color other than fluorescent pink, given that she gets the fluorescent pink one, we can use counting techniques.

First, we need to find the total number of ways Suzan can choose four marbles out of the 14 in the bag. This can be calculated using combinations, which is 14 choose 4 or (14!)/(4!10!) = 1001.

Next, we need to find the number of ways Suzan can choose one of each color other than fluorescent pink, given that she already picked the fluorescent pink one. There are three red, three green, three yellow, and four orange marbles left in the bag. Suzan needs to choose one from each color, which can be done in (3x3x3x4) = 108 ways.

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For the data generated by the questionnaire item that asked students to indicate their class rank in college, the appropriate measure of location would be the mode.

The mode is the value that occurs most frequently in a dataset and represents the most common response. In this case, the mode would indicate the most common class rank among the students surveyed. It is important to note that the use of the mode as a measure of location is most appropriate when dealing with nominal or ordinal data, such as class rank, where there is no inherent numerical relationship between the categories.

Other measures of location, such as the mean or median, are more appropriate for interval or ratio data where there is a meaningful numerical relationship between the values.

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The  absolute maximum value of f(x) on the interval [-3,9] is approximately 1.3 x 10^9 and the absolute minimum value of f(x) on the interval [-3,9] is 0.

To find the absolute maximum and minimum values of the function f(x) = x^8e^(-x) on the interval [-3, 9], we first need to find the critical points and endpoints of the function on the interval.

Taking the derivative of the function, we get:

f'(x) = x^7e^(-x)(8-x)

Setting f'(x) equal to zero, we get critical points at x=0 and x=8. We also need to check the endpoints of the interval, x=-3 and x=9.

Now we need to evaluate the function at these points to find the absolute maximum and minimum values.

f(-3) ≈ 3.3 x 10^5
f(0) = 0
f(8) ≈ 1.3 x 10^9
f(9) ≈ 4.4 x 10^8

Therefore, the absolute maximum value of f(x) on the interval [-3,9] is approximately 1.3 x 10^9 and the absolute minimum value of f(x) on the interval [-3,9] is 0.

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If 400 hypothesis tests are conducted with a null hypothesis assumed to be true and using a significance level of 1%, approximately 4 tests will incorrectly find significance. This is because 1% of 400 is 4 (0.01 x 400 = 4).

If the null hypothesis is true, then we would expect approximately 1% of the tests to result in a Type I error, which is incorrectly rejecting the null hypothesis. Therefore, out of the 400 tests conducted at a significance level of 1%, we would expect approximately 4 tests to incorrectly find significance.

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The estimate of the standard error of the mean (SE) is approximately 0.359 lb.

The standard error of the mean (SE) is a measure of the precision of the sample mean as an estimate of the population mean. It tells us how much variability there is in the sample means that we would expect if we took many random samples from the same population.

The formula for the SE is the standard deviation of the sample divided by the square root of the sample size.

In this case, we are given the sample mean = 10.85 lb, the sample standard deviation (s) = 4.25 lb, and the sample size (n) = 140 cats.

Using the formula, SE = s/√n, we can calculate the estimate of the SE to be:

SE = 4.25/√140 ≈ 0.359 lb

Therefore, we can estimate that the standard error is 0.359 lb.

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A researcher collected data on the age, in years, and the growth of sea turtles. The following graph is a residual plot of the regression of growth versus age. No, the residual plot does not support the appropriateness of a linear model because the graph displays a U -shaped pattern.

Researchers are employed in practically every industry or are paid to find, examine, and interpret data as well as identify patterns. They are employed in a variety of industries, including academics, science, medicine, and finance. Their workload is influenced by and dependent on their research objectives.

Through the use of the internet, books, articles in the press, surveys, and interviews, they develop information and collect data. No, the residual plot does not support the appropriateness of a linear model because the graph displays a U -shaped pattern.

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The Shavers Fork watershed is impacted by acid precipitation or other forms of pollution that are affecting the pH levels of the tributary water basin in the springtime.The data of Exercise 17.8 do give good reason to think that the springtime water in the tributary water basin around the Shavers Fork watershed is not neutral. Here's why:

Step 1: State the null hypothesis and the alternative hypothesis.

Null hypothesis: The springtime water in the tributary water basin around the Shavers Fork watershed is neutral (pH 7.0).

Alternative hypothesis: The springtime water in the tributary water basin around the Shavers Fork watershed is not neutral (pH ≠ 7.0).

Step 2: Determine the appropriate test statistic and the level of significance.

In this case, we can use a t-test for a single sample since we are comparing the pH of the springtime water in the tributary water basin to a neutral pH of 7.0. The level of significance is not given in Exercise 17.8, so we will assume it to be 0.05.

Step 3: Calculate the test statistic and the p-value.

Using the data from Exercise 17.8, we find that the sample mean pH is 6.45 and the sample standard deviation is 0.23. The test statistic is calculated as:

t = (6.45 - 7.0) / (0.23 / sqrt(9)) = -9.78

Using a t-table with 8 degrees of freedom (n-1), we find that the p-value is less than 0.001.

Step 4: Make a decision and interpret the results.

Since the p-value is less than the level of significance of 0.05, we reject the null hypothesis and conclude that the springtime water in the tributary water basin around the Shavers Fork watershed is not neutral. The data suggest that the pH of the water is significantly lower than a neutral pH of 7.0, indicating that the water is acidic. Therefore, we can infer that the Shavers Fork watershed is impacted by acid precipitation or other forms of pollution that are affecting the pH levels of the tributary water basin in the springtime.

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The rate of decay, denoted as 'r,' is a key factor in exponential functions, particularly when modeling real-world scenarios such as population decrease, radioactive decay, and depreciation of assets.

In an exponential decay function, the form is y = ab^(rt), where 'y' represents the remaining quantity, 'a' is the initial quantity, 'b' is the base, 'r' is the rate of decay expressed as a decimal, and 't' is the time elapsed.

The rate of decay, r, is a constant value that determines how rapidly the quantity decreases over time. It is expressed as a decimal (e.g., 0.2 for a 20% decay rate) and should be between 0 and 1 for decay scenarios. When determining the rate of decay, it is essential to gather data points that can be plotted on a graph to create an exponential curve, allowing you to estimate the decay rate accurately.

In some cases, the exponential decay equation can be written as y = ae^(-rt), where 'e' is the base of natural logarithms, approximately equal to 2.718. This is another representation of the same decay process and follows the same principles in terms of decay rate calculations.

To find the rate of decay for a specific dataset, you can use various techniques, including curve fitting or regression analysis, which help to find the best match between the data points and the exponential decay function.

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The df value for Monica's analysis is 98.

To calculate the degrees of freedom (df) for the Pearson correlation

coefficient, we need to know the sample size, which is the number of

observations or pairs of scores.

In this case, the sample size is 100 because Monica collected data from

100 people.

The formula for calculating the degrees of freedom for the Pearson

correlation coefficient is:

df = n - 2

where n is the sample size.

So, in this case, the degrees of freedom (df) would be:

df = 100 - 2 = 98

Therefore, the df value for Monica's analysis is 98.

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Answer:

-14°C

Step-by-step explanation:

The temperature at 12 noon = 10°C (given)

The temperature decreases by 2°C in 1 hour (given)

Thus, the temperature decreases by 1°C in 1/2 hour

Temperature 10°C above zero - Temperature 8°C below zero = 10 - (- 8) = 10 + 8 = 18°C

The temperature decreases by 18°C in 1/2 × 18 = 9 hours

Thus, from 10°C above zero to 8°C below zero it takes 9 hours

Total time = 12 noon + 9 hours

= 21 hours = 9 pm

Thus, at 9 pm temperature would be 8°C below zero.

(ii) The temperature at 12 noon = 10°C

The temperature decreases by 2°C every hour

The temperature decrease in 12 hours = - 2°C × 12 = - 24°C

At midnight, the temperature will be = 10°C + (-24°C) = -14 °C

Therefore, the temperature at mid night will be 14°C below 0.

To find the measure of angle BOC, set the expressions for m∠AOD and m∠BOC equal and solve for x. Then substitute the value of x back into the expression for m∠BOC to find its measure, which is 30°.

To find the measure of angle BOC, we can set the expressions for m∠AOD and m∠BOC equal to each other and solve for x.

7x - 5 = 3x + 15

Subtract 3x from both sides: 4x - 5 = 15

Add 5 to both sides: 4x = 20

Divide both sides by 4: x = 5

Now that we know x = 5, we can substitute it back into the expression for m∠BOC to find its measure.

m∠BOC = (3x + 15)° = (3*5 + 15)° = 30°

Therefore, the measure of ∠BOC is 30°, which corresponds to option B.

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Thus, there are approximately 408 students who are taking an online course and prefer basketball to football.

To solve this problem, we need to use the formula for the intersection of two independent events:

P(A and B) = P(A) * P(B)

where P(A) is the probability of event A occurring, P(B) is the probability of event B occurring, and P(A and B) is the probability of both events A and B occurring simultaneously.

In this case, let A be the event of taking an online class, and let B be the event of preferring basketball to football. We are asked to find the number of students who are in the intersection of these two events, or P(A and B).

We are given that there are 1200 students taking an online class, out of a total of 5000 students. Therefore, the probability of taking an online class is:

P(A) = 1200/5000 = 0.24

We are also given that 1700 students prefer basketball to football. Since this event is independent of taking an online class, the probability of preferring basketball to football is simply:

P(B) = 1700/5000 = 0.34

Now we can use the formula to find the probability of both events occurring simultaneously:

P(A and B) = P(A) * P(B) = 0.24 * 0.34 = 0.0816

Finally, we can convert this probability to a number of students by multiplying by the total number of students:

0.0816 * 5000 = 408

Therefore, there are approximately 408 students who are taking an online course and prefer basketball to football.

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Thus, the ladder reaches up approximately 11.28 feet of length up the side of the house.

The ladder represents the hypotenuse of a right triangle, the distance up the side of the house represents the opposite side, and the distance from the base of the ladder to the house represents the adjacent side.

Now, we can use the trigonometric function sine to find the length of the opposite side.

sin(70) = opposite/hypotenuse

sin(70) = x/12

x = 12sin(70)

Using a calculator, we can find that sin(70) is approximately 0.9397.

x = 12(0.9397)

x = 11.2764

Therefore, the ladder reaches up approximately 11.28 feet up the side of the house.

In summary, to find the distance up the side of the house that the ladder reaches, we used trigonometry and the sine function. The long answer to this problem explains the steps in detail and provides the numerical solution.

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To the nearest tenth, the approximate volume of a roll of pennies is 21650.0 mm³.

To find the approximate volume of a roll of pennies, we'll first calculate the volume of a single penny and then multiply that by the number of pennies in a roll (usually 50).

We will use the terms diameter and thickness in the calculation. Here are the steps:
Calculate the radius of the penny by dividing the diameter by 2:
Radius = Diameter / 2

= 19.05 mm / 2

= 9.525 mm.

Calculate the volume of a single penny using the formula for the volume of a cylinder (Volume = π × Radius² × Thickness):
Volume = π × (9.525 mm)² × 1.52 mm ≈ 433.0 mm³
Calculate the volume of a roll of 50 pennies:
Roll Volume = Single Penny Volume × Number of Pennies.

= 433.0 mm³ × 50 ≈ 21650 mm³.

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If the sand falls at a rate of 16 cubic millimeters per second, a player would have approximately 6.25 seconds for their turn.

The rate of sand falling from the hourglass is given as 16 cubic millimeters per second. We need to find out the time available for a turn. Let's assume that the hourglass is filled with 'x' cubic millimeters of sand.

We can use the formula:

Volume = Rate x Time

Here, the volume of sand is 'x' cubic millimeters, the rate is 16 cubic millimeters per second, and we need to find the time available for a turn, which we can represent as 't' seconds.

So,

x = 16t

We can rearrange this equation to find 't':

t = x/16

This means that the time available for a turn is equal to the volume of sand in the hourglass divided by the rate at which the sand falls.

We don't know the exact volume of sand in the hourglass, but let's assume it's 100 cubic millimeters.

Then,

t = 100/16

t = 6.25 seconds

So, in this case, a player would have approximately 6.25 seconds for their turn before all of the sand falls to the bottom of the hourglass.

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The gradient of the river is approximately 0.3947%, calculated by dividing the change in elevation of 1500 feet by the horizontal distance of 72 miles (converted to 380,160 feet).

The gradient of a river is the change in elevation divided by the horizontal distance.

Given that the change of elevation is 1500ft and the length of the river is 72 miles, we first need to convert the units to a consistent system. Let's convert the length from miles to feet, since the change in elevation is given in feet

72 miles = 72 x 5280 feet

72 miles = 380,160 feet

Now we can calculate the gradient using the formula

gradient = change in elevation / horizontal distance

gradient = 1500 ft / 380,160 ft

Simplifying, we get

gradient = 0.003947

Therefore, the gradient of the river is approximately 0.003947, which can be expressed as a percentage by multiplying by 100

gradient = 0.003947 * 100

gradient = 0.3947%

So, the gradient of the river is approximately 0.3947%.

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The maximum clearance for the bridge is 16 ft using the formula for parabolic arch.

Parabolic arches are well-known for their superior strength compared to other types of arches. These arches distribute weight more evenly, reducing the amount of stress that is placed on any given point. As a result, parabolic arches are commonly used in the construction of bridges.

In the case of a bridge with a supporting parabolic arch spanning 60 ft with a 30 ft wide road passing underneath, the minimum clearance needed is 16 ft. This means that the maximum height of the arch can be calculated by subtracting 16 ft from the height of the bridge.

To find the height of the bridge, we need to consider the formula for a parabolic arch: y = ax^2. Here, y is the height of the arch at any given point, x is the distance from the center of the arch, and a is a constant. The value of a will depend on the specific dimensions and properties of the arch.

In this case, we know that the span of the arch is 60 ft, so x = 30 ft at the halfway point. We also know that the height of the arch at this point is 16 ft + y, which gives us the equation [tex]16 ft + y = a(30 ft)^2[/tex].

Simplifying this equation, we get:

16 ft + y = 900a

To find the maximum clearance, we need to solve for y when a is at its maximum value. We can do this by finding the maximum value of a, which occurs at the apex of the arch.

The apex of a parabolic arch is located at the halfway point of the span, so x = 30 ft. We also know that the height of the arch at this point is 0 (since it is the highest point of the arch), so we can substitute these values into the formula to find the value of a:

[tex]0 = a(30 ft)^2[/tex]
a = 0

Now that we know that a = 0 at the apex of the arch, we can substitute this value into the equation we derived earlier:

16 ft + y = 900a
16 ft + y = 0
y = -16 ft

This means that the maximum clearance for the bridge is 16 ft, since the height of the arch at the apex is -16 ft (or 16 ft below the height of the bridge).

In conclusion, parabolic arches are known for their superior strength and are commonly used in bridge construction. By using the formula for a parabolic arch, we can calculate the maximum clearance for a bridge with a supporting parabolic arch. In this case, the maximum clearance is 16 ft, since the height of the arch at the apex is -16 ft.

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The order approximately 1.1111 cubic yards of topsoil from the dirt store.

The volume of topsoil required in cubic feet.

The area of the garden is:

The area of the garden is found by multiplying the length and width of the garden. In this case, the garden is 15 feet by 12 feet, so the area is 15 ft x 12 ft = 180 sq ft.

Since we want a 2 inch thick layer of topsoil, we need to convert the thickness to feet:

2 inches = 2/12 feet = 0.1667 feet

The volume of topsoil required in cubic feet is therefore:

180 sq ft × 0.1667 ft = 30 cubic feet

To convert this to cubic yards, we divide by 27 (since there are 27 cubic feet in a cubic yard):

The dirt store sells topsoil by the cubic yard, so we need to convert our answer from cubic feet to cubic yards. Since there are 27 cubic feet in a cubic yard (3 feet x 3 feet x 3 feet)

30 cubic feet ÷ 27 = 1.1111 cubic yards

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The value of c depends on the initial conditions x0 and y0. For example, if we are given x0 = 1 and y0 = 2, then c = 1. If we are given x0 = 2 and y0 = 5, then c = 8/3. To solve this problem, we first need to find the derivative of y with respect to x. Using the quotient rule, we get:

y' = (2x^(-3))(cx^4 - 2)

Next, we substitute y and y' into the differential equation and simplify:

xy' + 2y = 4x^2
x(2x^(-3))(cx^4 - 2) + 2(x^2 + c/x^2) = 4x^2
2cx - 2x^(-2) + 2x^2 + 2c/x^2 = 4x^2
2cx + 2c/x^2 = 6x^2
2c(x^3 + 1) = 6x^4
c = 3x/(x^3 + 1)

To satisfy the initial condition, we need to find the value of c that makes y(x) equal to some given value y0 when x = x0. Plugging in x0 and y0 into the equation for y, we get:

y0 = x0^2 + c/x0^2
c = x0^2(y0 - x0^2/x0^2)

In summary, the value of c that satisfies the given differential equation and initial condition depends on the specific values of x0 and y0. We can find c by plugging in these values into the equation for y and solving for c.

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We can say with 90% confidence that the increase in proportion of people satisfied with their financial situation is between 1.05% and 6.71%.

To calculate the confidence interval for the increase in proportion of people satisfied with their financial situation, we need to first calculate the proportions for both years:

Proportion in year 1 = 478/2038 = 0.2342
Proportion in year 2 = 537/1967 = 0.2730

The increase in proportion is:
0.2730 - 0.2342 = 0.0388

To calculate the confidence interval, we can use the formula:
CI = (point estimate ± (critical value x standard error))

The point estimate is the increase in proportion we just calculated: 0.0388

The critical value can be found using a z-table for a 90% confidence level. The z-value for a 90% confidence level is 1.645.

The standard error can be calculated using the formula:
sqrt[(p1(1-p1)/n1) + (p2(1-p2)/n2)]

where p1 and n1 are the proportion and sample size for year 1, and p2 and n2 are the proportion and sample size for year 2.

Plugging in the values, we get:
SE = sqrt[(0.2342(1-0.2342)/2038) + (0.2730(1-0.2730)/1967)] = 0.0174

Now we can plug in all the values to get the confidence interval:
CI = (0.0388 ± (1.645 x 0.0174)) = (0.0105, 0.0671)

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To compute the 99% confidence interval for the population mean, you need to determine the appropriate sample size to ensure that the sample mean does not deviate from the population mean by more than 1.3. The key terms involved in this process are the confidence interval, sample size, population mean, and sample mean.

The confidence interval represents the range within which the population parameter (in this case, the population mean) is likely to fall, given a certain level of confidence. A 99% confidence interval means that you are 99% confident that the true population mean falls within the specified range.

To calculate the required sample size, you will need to use the formula for the margin of error (E), which is E = (Zα/2 * σ) / √n, where Zα/2 is the critical value associated with the desired level of confidence (99%), σ is the population standard deviation, and n is the sample size.

Since you want the sample mean to not deviate from the population mean by more than 1.3, you will need to set E = 1.3 and solve for n. After finding the critical value for a 99% confidence interval (which is approximately 2.576) and assuming you know the population standard deviation, you can plug these values into the formula and solve for n.

By doing this, you will be able to determine the appropriate sample size to ensure that the 99% confidence interval for the population mean is within 1.3 units of the sample mean.

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The absolute minimum value of f on the given interval is 8+4π and the absolute maximum value of f on the given interval is 15.

To find the absolute maximum and absolute minimum values of f on the given interval [0,15], we need to first find the critical points of the function and then evaluate the function at those critical points as well as at the endpoints of the interval.

To find the critical points of f, we need to find the values of x where f'(x) = 0 or f'(x) does not exist.

f'(x) = 3x^2 - 1 - 25 = 3x^2 - 26

Setting f'(x) = 0, we get:

3x^2 - 26 = 0

x^2 = 26/3

x = ± √(26/3)

Since √(26/3) is not in the interval [0,15], we only need to consider x = - √(26/3) as a critical point.

Next, we evaluate the function f at the critical point and at the endpoints of the interval:

f(0) = 0(0)^2 - 0 - 25 = -25

f(15) = 15(15)^2 - 15 - 25 = 3375 - 15 - 25 = 3335

f(-√(26/3)) = (-√(26/3))(√(26/3))^2 - (-√(26/3)) - 25

= 26/3 + √(26/3) - 25

To compare these values and find the absolute maximum and minimum, we can use the following observations:

- If the critical point or an endpoint gives the largest value of f, then that is the absolute maximum.
- If the critical point or an endpoint gives the smallest value of f, then that is the absolute minimum.

Comparing the values we found, we can see that:

- The absolute minimum value of f on [0,15] is 26/3 + √(26/3) - 25 ≈ 8 + 4π
- The absolute maximum value of f on [0,15] is 15

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Based on the information given in (6), we know that the mean is 65 and the standard deviation is 3. we can use the empirical rule to estimate the probability of the sample mean being within one standard deviation away from the mean in either the positive or negative direction.

According to the empirical rule, approximately 68% of the sample means will fall within one standard deviation away from the mean in either direction. Therefore, the probability of the sample mean being within one standard deviation away from the mean in either the positive or negative direction is approximately 0.68 or 68%.

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The total amount owed by the students for the week is $[tex]$28[/tex].

An amount owed refers to total of the money a person owe us from time to time which can include loan and any unpaid interest, fees and expenses. It can also be an ordinal expenses such as school fee etc.

The total amount owed by the student will be:

= $3.00 + $8.00 + $7.00 + $10.00

= $28

Full question "On the first Monday of each month, the school sends home a note that includes each student's lunch account balance. These students owe money. Student 1 =$3.00, Student 2 - $8.00, Student 3 - $7.00, Student 4 - $10.00. What is the total amount owed by the student?".

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The probability that the return of the portfolio will be less than -43.5% in any given year is 0.0139, or approximately 1.39%.

To solve this problem, we need to standardize the value of -43.5% using the given mean and standard deviation.

z = (x - mu) / sigma

where z is the z-score, x is the value we want to find the probability for (-43.5%), mu is the expected return (13.2%), and sigma is the standard deviation (18.9%).

Substituting the given values:

z = (-0.435 - 0.132) / 0.189

z = -2.22

We can use a standard normal distribution table or calculator to find the probability that a standard normal random variable is less than -2.22.

P(Z < -2.22) = 0.0139

Therefore, the probability that the return of the portfolio will be less than -43.5% in any given year is 0.0139, or approximately 1.39%.

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The amount of milk Jonah bring is 8 quarts of milk

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

Jonah brought 16 pints of milk to share with his soccer teammates at halftime.

This means that

Milk = 16 pints of milk

By the metric units of conversion, we have

1 pint of milk = 0.5 quart of milk

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

Milk = 16 quarts of milk * 0.5

Evaluate

Milk = 8 quarts of milk

Hence, the amount of milk is 8 quarts of milk

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Answer:

22/30

33/45

7.33333...

%73.33..

Step-by-step explanation:

to find it in a different form as a fraction you can simply multiply it by any number greater than 1 and just make sure you multiply the numerator and the denominator by the same number

to get a decimal simply divide it

to get a percentage you divide the numbers and multiply that by 10.

The answer is B, which is to find the common denominator.

In solving a rational equation, the first step is usually to find a common denominator for all the fractions in the equation. This allows us to combine the fractions into a single fraction and simplify the equation.

To find the common denominator, we need to identify the factors of the denominators and determine the least common multiple (LCM) of these factors. Then, we multiply each fraction by the appropriate factor(s) to get the common denominator. For example, if we have the equation:

(3/x) + (4/2x) = 1/4

The denominators are x and 2x, which have factors of x and 2. The LCM of these factors is 2x, so we need to multiply the first fraction by 2 and the second fraction by 1 to get:

(6/2x) + (4/2x) = 1/4

Then we can combine the fractions and simplify the equation to get:

(10/2x) = 1/4

From here, we can continue to solve for x by cross multiplying and manipulating the equation.

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[tex]\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ S = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\[-0.5em] \hrulefill\\ n=4 \end{cases}\implies S=180(4-2)\implies S=360[/tex]

so since a quadrilateral will have a total of 360°, minus 240 and 20 that leaves us with only 100° leftover, now to make it in a 3 : 7 ratio, let's simply divide 100 by (3 + 7) and distribute accordingly.

[tex]3~~ : ~~7\implies 3\cdot \frac{100}{3+7}~~ : ~~7\cdot \frac{100}{3+7}\implies 3\cdot 10~~ : ~~7\cdot 10\implies 30^o~~ : ~~70^o[/tex]

Answer:

Step-by-step explanation:

You want to know the number of 5-letter combinations of 3 consonants and 2 vowels can be formed from the letters of "FORMULATED", and the number that have a vowel at each end.

Since the problem statement uses the word "combinations" instead of "permutations", we take it to mean that "FORMU" is to be considered the same as "FOMRU" and "FUMRO", which have the same letters.

Since the position of the vowels seems to matter, either of the above is considered different from "FORUM" where the vowels are in different places.

The number of combinations of 3 consonants from the 6 in "FORMULATED" is 6C3 = 6!/(3!(6 -3)!) = 20, and the number of combinations of 2 vowels of the 4 given is 4C2 = 4!/(2!(4-2)!) = 6.

The possible arrangements of 2 vowels and 3 consonants in a group of 5 letters is 5C2 = 5!/(2!(5-2)!) = 10.

So, the combinations of 3 consonants and 2 vowels in with vowels in the different possible positions is ...

  20·6·10 = 1200

There can be 1200 different 5-letter combinations.

There is only one of the 10 possible arrangements of consonants and vowels that has the vowels at each end. The number of such arrangements is 1/10 of the total, or (1/10)(1200) = 120.

120 of the letter combinations will have a vowel at each end.

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Additional comment

More often, we're interested in the number of "words", where letter order matters. If that is intended to be the case, then the number of 5-letter "words" is 6P3·4P2·5C2 = 14400, and the number that have vowels at each end is 1440.

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