1. If Cos A = 1/4, what is Sin B?

2. Simplify Sin A + Cos B ÷ 2

3. If Tan B = 1/5, what is Tan A? Which angle is bigger, angle A or angle B?

Please answer all 3

The pigeon should fly directly from point C to point B, then fly along the shoreline to a point 10.387 miles away from point A (rounded to three decimal places). This can be found using the principle of minimizing the total distance traveled, taking into account the different energy rates over water and land.

To minimize the total energy required for the homing pigeon to get to its home loft at Point A, we need to find the optimal point downshore, Point S, to fly to. Using the given information, we can set up a function for the total energy.

Let x be the distance from Point A to Point S. Then, the pigeon will fly x miles over land and the remaining distance, 20-x miles, downshore from Point B to Point S. The distance from Point C to Point S can be found using the Pythagorean theorem:

CS = sqrt((20-x)^2 + 9^2)

Since the pigeon uses energy at a rate 1.29 times over water compared to land, we can write the total energy function as:

E(x) = x + 1.29 * CS

Now we need to minimize this function. To do so, we can take the derivative of E(x) with respect to x and set it equal to zero:

dE(x)/dx = 0

By solving this equation for x, we will find the optimal distance downshore from Point A to Point S. Once you have the value of x, you can say that Point S is x miles away from Point A (rounded to three decimal places, as needed).

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A: An equation that represents the problem is 1.75x - 0.45 = 4.45. B: Solving the equation from Part A gives x = 2.8. C: The solution to the equation represents the number of pounds of apple bought by Casey.

Part A: Write an equation that represents the problem. Define any variables.

Let x represent the number of pounds of apples Casey bought. The cost of apples is $1.75 per pound, so the total cost before using the coupon would be 1.75x. After using the $0.45 coupon, her total was $4.45. The equation representing this situation is:

1.75x - 0.45 = 4.45

Part B: Solve the equation from Part A.

Now, let's solve the equation:

1.75x - 0.45 = 4.45

Add 0.45 to both sides:

1.75x = 4.90

Now, divide both sides by 1.75:

x = 4.90 / 1.75

x = 2.8

Part C: Explain what the solution to the equation represents

The solution, x = 2.8, represents that Casey bought 2.8 pounds of apples at the grocery store.

Note: The question is incomplete. The complete question probably is: Apples are on sale at a grocery store for $1.75 per pound. Casey bought apples and used a coupon for $0.45 off her purchase. Her total was $4.45. How many pounds of apples did Casey buy? Part A: Write an equation that represents the problem. Define any variables. Part B: Solve the equation from Part A. Show all work. Part C: Explain what the solution to the equation represents.

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Answer:

Given the equation of the circle in the xy-coordinate plane as x^2 + y^2 + 6x - 4 = 0, we need to rewrite it in the form (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.

Completing the square for x^2 + 6x, we get (x+3)^2 - 9. Thus, the equation becomes (x+3)^2 + y^2 - 9 = 4. Rearranging, we get (x+3)^2 + y^2 = 13.

Comparing with the required form, we get (x-h)^2 + (y-k)^2 = r^2, where h = -3, k = 0 and r^2 = 13. Thus, the value of k is 0.

each of Jennifer's five kids received 1/40 of the total pan of macaroni and cheese.

What is equivalent ratio?

The concept of a ratio in mathematics is the divisional comparison of two quantities, the antecedent and consequent. As an illustration, each ingredient must be added according to a ratio during cooking. So, we may argue that a ratio is employed to represent one quantity as a portion of another. The ratio can be written as a fraction as well. If the ratio a:b is a fraction, its form is a/b. As a result, it is simple to compare two or more equivalent ratios expressed as equivalent fractions.

If Jennifer had 7/8 of her pan of macaroni and cheese left after supper, this means she had 1 - 7/8 = 1/8 of the pan remaining.

To split the remaining macaroni and cheese evenly between her five kids, we need to divide 1/8 by 5.

1/8 divided by 5 can be written as (1/8) ÷ 5 = (1/8) x (1/5) = 1/40.

Therefore, each of Jennifer's five kids received 1/40 of the total pan of macaroni and cheese.

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The volume of the prism is 21 * 5 = 105 cubic feet.

To find the volume of a prism with a triangular base, you need to follow these steps:

1. Determine the area of the triangular base: Since the base is a right triangle with leg lengths of 6 feet and 7 feet, you can use the formula for the area of a right triangle: (1/2) * base * height. In this case, the area would be (1/2) * 6 * 7 = 21 square feet.

2. Multiply the area of the triangular base by the height of the prism: The prism is 5 feet tall, so the volume can be calculated by multiplying the area of the base (21 square feet) by the height (5 feet).

Thus, the volume of the prism is 21 * 5 = 105 cubic feet.

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The opening of the pail has a diameter of approximately 55.4 inches (since it's twice the diameter of the base).

Based on the information you provided, it sounds like you have some great supplies for making sand castles at the beach with your friends.

Firstly, let's take a look at the pail. You mentioned that it has a base with a circumference of 87 inches, which means that the diameter of the base is approximately 27.7 inches (since circumference = pi x diameter). The pail is also 12 inches tall and has an opening on top that is twice the diameter of the base. Therefore, the opening has a diameter of approximately 55.4 inches (since it's twice the diameter of the base). With these measurements, you can use the pail to make some pretty big sand castles!

Next, you mentioned a plastic pyramid mold that has a square base with an edge that measures 6 inches and is 7 inches tall. This mold should be perfect for making pyramid-shaped sand castles. Just fill it with sand, pack it down, and carefully remove the mold to reveal your pyramid!

Finally, you mentioned an empty soup can with a diameter of 5.25 inches and is 6.5 inches tall. This can could be used to make cylindrical shapes in your sand castles. Simply pack sand around the can, press down firmly, and carefully remove the can to reveal your cylinder.

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Based on mathematical operations, since the cookies Raina wants for a party comes in packs of 6 and she needs 23 of these for the party, she should buy 4 packs.

The basic mathematical operations include addition, subtraction, multiplication, and division.

In this situation, we can determine the number of packs required by finding a number that when multiplied by 6 will be close to 23.

We know that 6 x 4 equals 24, which is the closest value to 23.

The number of cookies in each pack = 6

The total number of cookies Raina needs = 23

The number of packs to buy = 4 (6 x 4 = 24)

Thus, Raina needs to buy 4 packs of the cookies to satisfy her requirement for 23 pieces.

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Loan H's monthly payment will be $42.46 smaller than loan I's (rounded to the nearest cent). The correct option is a.

To determine which loan will have the smaller monthly payment, we need to calculate the monthly payments for both loans using the given information.

For loan H, the monthly interest rate is 12.24%/12 = 1.02%, and the number of payments is 3 years x 12 months/year = 36. Using the formula for the monthly payment on a loan with monthly compounding, we have:

P = (r(PV))/(1 - (1+r[tex])^{(-n)})[/tex]

where P is the monthly payment, r is the monthly interest rate, PV is the principal value of the loan, and n is the total number of payments.

Plugging in the values given for loan H, we get:

P = (0.0102 x $5,650) / (1 - (1+0.0102)⁻³⁶) = $186.25

For loan I, the monthly interest rate is 10.97%/12 = 0.9142%, and the number of payments is 4 years x 12 months/year = 48. Using the same formula as above, we have:

P = (0.009142 x $6,830) / (1 - (1+0.009142)⁻⁴⁸) = $227.04

Therefore, the monthly payment for loan H is $186.25 and the monthly payment for loan I is $227.04.

To find the difference between the monthly payments, we subtract the monthly payment for loan H from the monthly payment for loan I:

$227.04 - $186.25 = $40.79

Therefore, the answer is (a) loan H's monthly payment will be $42.46 smaller than loan I's (rounded to the nearest cent).

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The amount of interest Amy will pay over the 3 years is $105.

Simple interest is a method of calculating the interest amount on a loan or investment by multiplying the principal amount, the annual interest rate, and the time in years. In Amy's case, she borrowed $1,000 with an annual interest rate of 3.5% and will take 3 years to pay back the loan.

To calculate the interest Amy will pay, use the formula: Interest = Principal x Rate x Time

Interest = $1,000 x 0.035 (3.5% as a decimal) x 3 years

Interest = $1,000 x 0.035 x 3 = $105

Amy will pay $105 in interest over the 3 years.

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The value of the expression 65,536 x 4h for h = -4 is 256. The value represents the number of bacteria in the culture 4 hours before 8:00 A.M.

To find the value of the expression for h = -4, we substitute -4 for h in the expression [tex]65536*4^{h}[/tex]. This gives us

65536*4⁻⁴

To simplify this, we need to evaluate the exponent first. Remember that a negative exponent means we take the reciprocal of the base raised to the positive version of the exponent. So 4⁻⁴ is the same as 1/(4⁴), or 1/256.

Substituting that back into the expression, we get

65536*(1/256) = 256

So the value of the expression for h = -4 is 256.

The value of the expression in part (a) represents the number of bacteria in the culture 4 hours before 8:00 A.M. Since h is negative, we are looking at a time before 8:00 A.M. Specifically, h = -4 means we are looking at 4 hours before 8:00 A.M., or 4:00 A.M.

So the expression 65536*4⁻⁴ tells us how many bacteria were in the culture at 4:00 A.M., assuming the bacteria quadrupled every hour from that point until 8:00 A.M.

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The derivative of the given function is:

y' = (8x – 9) e⁻⁴ - 16x (4x - 9) e⁻⁴

Find the derivative?

To find the derivative of the given function y= (4x² – 9x) e⁻⁴, we need to use the product rule of differentiation. The formula for the product rule is:

(fg)' = f'g + fg'

Where f and g are two differentiable functions. Applying this formula, we get:

y' = (4x² – 9x)' e⁻⁴ + (4x² – 9x) (e⁻⁴)'

The first term on the right-hand side can be simplified using the power rule and the constant multiple rule of differentiation:

(4x² – 9x)' = 8x – 9

The second term on the right-hand side requires the chain rule of differentiation. Let u = -4x, then we have:

(e⁻⁴)' = (e^u)' = e^u (-4) = -4e⁻⁴x

Substituting these results back into the expression for y', we get:

y' = (8x – 9) e⁻⁴ + (4x² – 9x) (-4e⁻⁴x)

Simplifying this expression, we get:

y' = (8x – 9) e⁻⁴ - 16x (4x - 9) e⁻⁴

Therefore, the derivative of the given function is:

y' = (8x – 9) e⁻⁴ - 16x (4x - 9) e⁻⁴

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A frequency table was created using data points representing the number of holes in each of Grandma Marion's dresses. The table shows the number of dresses with 5, 7, and 8 holes, respectively.

To create a frequency table for the given data, first, the unique values in the data set are identified, which are 5, 7, and 8. Then, the number of occurrences of each unique value is counted, resulting in the frequencies 1 for 5, 2 for 7, and 3 for 8.

Count the frequency of each data point

5: 1 dress

7: 2 dresses

8: 3 dresses

Finally, these values are organized into a table with two columns, one for the unique values and another for their corresponding frequencies. The resulting frequency table shows the number of dresses with each number of holes eaten by moths.

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The maximum profit will be realized by producing and selling 2.5 kilograms of whatchamacallits.

The maximum profit, in hundreds of dollars, will be $500.

To find the maximum profit, we need to first calculate the profit function P(x), which is the difference between the revenue and the cost functions:

P(x) = R(x) - C(x)
P(x) = 450x - (50x + 1000)
P(x) = 400x - 1000

To find the amount of kilograms that should be produced and sold to realize a maximum profit, we need to find the value of x that maximizes the profit function.

We can do this by taking the derivative of the profit function and setting it equal to zero:

P'(x) = 400
400 = 0

Since the derivative is a constant value, there is no critical point or inflection point. Therefore, the profit function is increasing at a constant rate, and the maximum profit will be achieved at the highest possible value of x.

To find that value, we can set the profit function equal to zero and solve for x:

P(x) = 400x - 1000 = 0
400x = 1000
x = 2.5

Therefore, the maximum profit will be realized by producing and selling 2.5 kilograms of whatchamacallits.

To find the maximum profit itself, we can substitute this value of x into the profit function:

P(2.5) = 400(2.5) - 1000 = 500

So the maximum profit, in hundreds of dollars, will be $500.

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Answer: 832 plants

Step-by-step explanation:

If there are 4 plants for every 1 square ft the ratio is 4:1.

This tells us to multiply 208x4 giving us 832.

According to recent studies, there is a notable difference in credit scores between those over the age of 55 and those between the ages of 18-24.

On average, individuals over the age of 55 tend to have higher credit scores than those in the 18-24 age range. This is primarily due to the fact that older individuals have had more time to establish and build their credit history, whereas younger individuals are just starting out and may not have had the opportunity to establish credit yet.

Additionally, older individuals tend to have more stable financial situations and may have less debt compared to younger individuals who may be dealing with student loans or other types of debt.

However, it's important to note that credit scores can vary greatly between individuals, regardless of age, and there are many factors that contribute to credit score, such as payment history, credit utilization, and length of credit history.

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With the surface area of the square pyramid 84 square inches and side length of the base is 6, the value of x is 4 inches, by assuming x as the slant height of the square pyramid.

Assuming that x refers to the slant height of the square pyramid, we can use the formula for the surface area of a square pyramid to solve for x:

Surface area of a square pyramid = base area + (0.5 x perimeter of base x slant height)

Since the base of the square pyramid is a square with side length 6,

the base area is 6² = 36 square inches.

The perimeter of the base is 4 times the side length, so it is 4 x 6 = 24 inches.

Substituting these values into the formula and simplifying, we get:

84 = 36 + (0.5 x 24 x x)

84 - 36 = 12x

48 = 12x

x = 4

Therefore, the value of x, the slant height of the square pyramid, is 4 inches.

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Answer:

(x-6)²+(y+3)²=17²

Step-by-step explanation:

The equation of a circle with center (a,b) and radius r is given by the formula:

(x - a)² + (y - b)² = r²

In this case, the center of the circle is (6,-3), and it passes through the point (-9,5). To find the radius of the circle, we need to calculate the distance between the center and the point on the circle. Using the distance formula, we get:

r = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(-9 - 6)² + (5 - (-3))²]

= √[225 + 64]

= √289

= 17

So, the radius of the circle is 17. Now we can plug in the values for the center and radius into the equation of a circle:

(x - 6)² + (y + 3)² = 17²

Answer:

The correct answer is D. The relationship is not a direct proportion.

We can see that the money raised is not directly proportional to the number of cars washed. For example, when the number of cars washed is doubled from 3 to 13, the money raised is not doubled from $43.50 to $87.00. Instead, it is increased by a factor of 6.5, from $43.50 to $279.50. Similarly, when the number of cars washed is increased by 5 from 13 to 18, the money raised is increased by a factor of 1.4, from $279.50 to $405.00.

This suggests that the amount of money raised is not simply a linear function of the number of cars washed. Instead, it is likely a more complex function that takes into account other factors, such as the time of day, the weather, and the location of the car wash.:

Step-by-step explanation:

C= 2πr

Given,

Diameter= 9

so, radius = 9÷2 = 4.5

C= 2 x π x 4.5

= 28.3 (3.s.f)

The value of measure of arc QI is,

⇒ m QI = 94°

We have to given that;

⇒ m YS = 180°

⇒ m ∠QBI = 137°

Hence, We can formulate;

⇒ m ∠QBI = 1/2 (m YS + m QI)

⇒ 137 = 1/2 (180 + m QI)

⇒ 274 = 180 + m QI

⇒ m QI = 274 - 180

⇒ m QI = 94°

Thus, The value of measure of arc QI is,

⇒ m QI = 94°

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The percent discount on the phone because of a sale is 45%.

To find the percent discount, we need to calculate the difference between the original price and the discounted price, and then express that difference as a percentage of the original price.

First, let's find the difference between the two prices: $28 - $15.40 = $12.60. This means that Benjamin saved $12.60 on the phone.

Now, let's find the percent discount. We can do this by dividing the savings by the original price, and then multiplying the result by 100: ($12.60 / $28) * 100 = 45%.

So, Benjamin received a 45% discount on the phone before tax. This calculation shows that the sale allowed him to save a significant amount on his purchase. It's important to compare original and discounted prices to determine if a sale provides good value.

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she can add each number one by one , and whenever she needs to carry numbers over. She can add them to the already existing numbers. she cna check with a calculator

The volume of the rectangular prism is 200 in³.

The volume of a rectangular prism is found by multiplying its length, width, and height. In this case, the prism has a length of 8 inches, width of 2 inches, and height of 12 and one-half inches. To calculate the volume, you would use the following formula:

Volume = Length × Width × Height

Now, plug in the given dimensions:

Volume = 8 in × 2 in × 12.5 in

Perform the calculations:

Volume = 16 in² × 12.5 in

Volume = 200 in³

So, the rectangular prism has a volume of 200 cubic inches. This means that the space occupied by the prism is equal to 200 cubic inches. The other options provided, such as 16 in³, 22 and one-half in³, and 212 and one-half in³, are not correct because they do not represent the product of the length, width, and height of the given prism. In conclusion, the correct answer for the volume of this rectangular prism is 200 in³.

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Curtis started with 84 cards and now has 12 cards at home and 9 cards at school, for a total of 21 cards.

To find how many card ,We see Curtis has 9 cards at school after trading with Dino again, which means he had 12 cards before the trade.

Before his mom bought him another pack of 12 cards, he had 10 missing, so he must have had 24 cards in total (12 + 12).

He hid half of his cards at home, so he has 12 cards at home.

He traded ¼ of the cards he brought to school to Dino and got back 3 of Dino's cards. Let's call the number of cards he brought to school "x".

So, he traded x/4 cards to Dino, and got back 3 cards, which means he now has (x/4) - 3 cards.

We know that he now has 9 cards at school, so we can set up an equation:

(x/4) - 3 = 9

Solving for x, we get:

x/4 = 12

x = 48

So, Curtis brought 48 cards to school, which means he started with 24 + 12 + 48 = 84 cards in total.

Therefore, Curtis started with 84 cards and now has 12 cards at home and 9 cards at school, for a total of 21 cards.

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To find the arc length of the curve y = -x² -5 on the interval [-1,2], we use the formula to evaluate:

L = ∫√(1 + (dy/dx)²) dx

where dy/dx is the derivative of y with respect to x.

First, we find dy/dx:

dy/dx = -2x

Next, we substitute dy/dx into the formula and simplify:

L = ∫√(1 + (-2x)²) dx
L = ∫√(1 + 4x²) dx

To evaluate this integral, we can use a trigonometric substitution. Let x = (1/2)tanθ, then dx = (1/2)sec²θ dθ. Substituting, we get:

L = ∫√(1 + 4(1/2)²tan²θ)(1/2)sec²θ dθ
L = (1/2)∫sec³θ dθ

To integrate sec³θ, we use integration by parts:

u = secθ, du/dθ = secθ tanθ
dv/dθ = sec²θ, v = tanθ

∫sec³θ dθ = secθ tanθ - ∫tan²θ secθ dθ
= secθ tanθ - ∫secθ dθ + ∫sec³θ dθ

Rearranging, we get:

2∫sec³θ dθ = secθ tanθ + ln|secθ + tanθ|

Therefore:

L = (1/2)(secθ tanθ + ln|secθ + tanθ|) + C

To evaluate L on the interval [-1,2], we need to find θ when x = -1 and x = 2. Using the substitution x = (1/2)tanθ:

When x = -1, θ = -π/4
When x = 2, θ = π/3

Substituting these values into the equation for L and simplifying, we get:

L = (1/2)(2√2 + ln(3 + 2√3) + π/4)

Therefore, the integral that gives the arc length of the curve y = -x² -5 on the interval [-1,2] is:

L = (1/2)(2√2 + ln(3 + 2√3) + π/4)

Note: If technology is used to evaluate or approximate the integral, the answer may differ slightly due to rounding errors.

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The sum of the polynomials (5x³ + x) + (3x³ + 8) is 8x³ + x + 8

(5x³ + x) + (3x³ + 8) can be simplified by adding the coefficients of the like terms. The like terms are 5x³ and 3x³, which can be combined to give 8x³. The single terms are x and 8, which can be combined to give 8 + x. Therefore, the polynomials add up to

8x³ + x + 8

In standard form (highest exponent to lowest), the answer is:

8x³ + x + 8

Therefore, the sum of the polynomials (5x³ + x) + (3x³ + 8) is 8x³ + x + 8 in standard form (highest exponent to lowest)

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[tex] \sf \longrightarrow \: \frac{2n - 1}{3} = \frac{n + 2}{2} \\ [/tex]

[tex] \sf \longrightarrow \: 2( 2n - 1) = 3(n + 2) \\ [/tex]

[tex] \sf \longrightarrow \: 4n - 2 = 3n +6 \\ [/tex]

[tex] \sf \longrightarrow \: 4n = 3n +6 + 2\\ [/tex]

[tex] \sf \longrightarrow \: 4n - 3n= 6 + 2\\ [/tex]

[tex] \sf \longrightarrow \: 1n= 6 + 2\\ [/tex]

[tex] \sf \longrightarrow \: n= 8\\ [/tex]

[tex] \longrightarrow { \underline{ \overline{ \boxed{ \sf{\: \: \: n= 8 \: \: \: }}}}} \: \: \bigstar\\ [/tex]

The result of subtracting 5 from 3 1/3 is -2 2/3.

To subtract 5 from 3 1/3, we need to first convert the mixed number to an improper fraction. This can be done by multiplying the whole number (3) by the denominator of the fraction (3), and adding the numerator (1) to get 10/3. Therefore, 3 1/3 is equivalent to 10/3.

Next, we can subtract 5 from 10/3 by finding a common denominator of 3, which gives 15/3 - 10/3 = 5/3. This is the result in improper fraction form.

To convert back to a mixed number, we can divide the numerator (5) by the denominator (3), which gives a quotient of 1 and a remainder of 2. Therefore, the answer is -2 2/3.

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Where every student offer at least one of the subjects, there are 4 students who offer all three subjects.

To find the number of students who offer only one subject, we can use the formula:

number of students offering only one subject = total number of students offering the subject - number of students offering both subjects

For math, there are 22 students offering math and 5 offering both math and biology, so the number of students offering math only is:

22 - 5 = 17

Similarly, for physics and biology, the numbers of students offering the subjects only are:

14 - 7 = 7 (physics)
16 - 5 = 11 (biology)

To find the number of students offering all three subjects, we can use the formula:

number of students offering all three subjects = total number of students offering at least one subject - number of students offering only one subject in each subject + number of students offering no subject

We know that there are 7 students offering at least one subject. To find the number of students offering no subject, we can subtract this from the total number of students:

39 - 7 = 32

Now we can plug in the numbers:

number of students offering all three subjects = 7 - 17 - 7 - 11 + 32
= 4

Therefore, there are 4 students who offer all three subjects.

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