If a ball is thrown into the air with a velocity of 36 ft/s, its height (in feet) after t seconds is given by y = 36t − 16t2. Find the velocity when t = 1.

Answer:

To prove that there exist infinitely many integers n such that n^2 + 1 is squarefree, we can use a proof by contradiction.

Assume the contrary, that there are only finitely many integers n such that n^2 + 1 is squarefree. Let's denote these integers as n_1, n_2, ..., n_k, where k is a finite positive integer.

Consider the number N = (n_1^2 + 1) * (n_2^2 + 1) * ... * (n_k^2 + 1) + 1.

Note that N is an integer and is greater than each of the numbers n_i^2 + 1 for i = 1 to k. Since n_i^2 + 1 is squarefree for each i, N is also squarefree, as it is not divisible by any square number other than 1.

However, N cannot be equal to any of the n_i^2 + 1, as N is strictly greater than each of them. This contradicts our assumption that n_1, n_2, ..., n_k are all the integers such that n^2 + 1 is squarefree.

Therefore, our assumption must be false, and there exist infinitely many integers n such that n^2 + 1 is squarefree. This completes the proof by contradiction.

Step-by-step explanation:

A.) Area = (1/2) × 7 × 6 × √(15)/4 = 21√(15)/4

B.) Angle C is 75 degrees.

C.) Sin(A) is 3√(255)/85 in simplest radical form.

Trigonometry is a branch of mathematics that examines triangle-related relationships and calculations, particularly the measurement of triangle sides and angles.

A.) We can apply the following formula to determine the triangle's area:

Area = (1/2) × b × a × sin(C)

Since we know b=7 and a=6, we have:

Area = (1/2) × 7 × 6 × sin(C)

To find sin(C), we can use the fact that cos(C) = 1/4 and the Pythagorean identity:

sin²(C) + cos²(C) = 1

sin(C) = √(1 - cos²(C)) = √(1 - 1/16) = √(15/16) = √(15)/4

B.) To find C, we can use the Law of Cosines:

c² = a² + b² - 2ab×cos(C)

c² = 6² + 7² - 2(6)(7)(1/4) = 85/2

c = √(85)/2

Now we can use the Law of Cosines again to find angle C:

cos(C) = (a² + b² - c²)/(2ab)

cos(C) = (6² + 7² - (√(85)/2)²)/(267) = 1/4

C.) To find sin(A), we can use the Law of Sines:

sin(A)/a = sin(C)/c

sin(A) = asin(C)/c = 6(√(15)/4)/(√(85)/2) = 3×√(15)/√(85) = √(255)/85

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Answer:

Step-by-step explanation:

Q:

In a particular class of 29 students, 11 are men. What fraction of the students in the class are women?

A:

29-11=18

18/29 are women

Answer:

complementary since the angles sum up to 90

To find the volume of a ball, we use the formula V = (4/3)πr³, where r is the radius of the ball. We are given the diameter of the ball, which is 9 inches.

The radius of the ball is half of the diameter, so

r = 9/2 = 4.5 inches.

Substituting this value of r in the formula, we get:

V = (4/3)π(4.5)³V = (4/3)π(91.125)V = 4.1888 × 91.125V ≈ 381.71

Therefore, the volume of the ball is approximately 381.71 cubic inches.

When rounded to the nearest hundredth, the answer is 381.71.

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The solution of the expression is; |y-x| = y - x without using absolute value.

Numerical articulation is characterized as the assortment of the numbers factors and works by utilizing tasks like expansion, deduction, increase, and division.

Given that;

from the question the expression is;

⇒ |y−x| , if y > x

⇒ If y > x,

⇒ y - x > 0

then |y-x| is equal to (y-x),

since (y-x) is always positive in this case.

Therefore, The solution of the expression is; |y-x| = y - x

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Answer: The greatest distance a person could be from the sprinkler and get sprayed by it is 14 feet.

Step-by-step explanation:

The equation given describes a circle with the general equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

In the provided equation, (z + 6)^2 + (y - 9)^2 = 196, we can identify the values for h, k, and r^2:

1. h = -6 (from z + 6)

2. k = 9 (from y - 9)

3. r^2 = 196

Now, we need to find the radius r, which is the greatest distance a person could be from the sprinkler and still get sprayed by it. To do this, we take the square root of r^2:

r = √196

r = 14

The value of tetha is 330°

If a ray rotates in an anticlockwise direction, then the angle formed as a product of the rotation is called a positive angle.

In other words if it rotates in clockwise direction, it is a negative angles.

If sin(tetha) = -0.5

to find the value of tetha we multiply both sides with the inverse of sin

therefore;

tetha = sin^-1(-0.5)

tetha = -30

therefore -30 is a negative angle, to find the positive angle of tetha, we add 360

therefore tetha = -30+360

= 330°

therefore the value if tetha that will be greater than 279 but less than 360 is 330°

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Answer:

12 minutes

Step-by-step explanation:

11/x = 55/60

11 x 5 = 55

60/5=12

12.01 minutes

time = distance / speed

where distance is the distance traveled, speed is the average speed of the travel, and time is the time taken to cover the distance.

We are given that the distance traveled is 11 miles and the average speed is 55 miles per hour. To convert the speed to miles per minute, we can divide it by 60 since 1 hour is equal to 60 minutes:

55 miles per hour = 55/60 miles per minute

= 0.9167 miles per minute (rounded to four decimal places)

Now we can substitute the values into the formula:

time = distance / speed

= 11 miles / 0.9167 miles per minute

≈ 12.01 minutes (rounded to two decimal places)

Therefore, it would take approximately 12.01 minutes to travel 11 miles at an average speed of 55 miles per hour.

*IG:whis.sama_ent

Answer:

1/2

Step-by-step explanation:

Let's check if this is a positive of negative slope. Remember that when checking slope, you need to read the graph from left to right.

So let's see: When we look at the graph from left to right, is it moving up or down?

It is moving up, correct!

Therfore, this must be a positive slope.

Ok, now we need to pick 2 points.

I will choose: ( 6 , 0 ) & ( 2 , -2 )

To move between those pouints, we need to move up 2 and 4 units to the right.

Slope is defined as:

Change is y over change in x

Change in y is positive 2

Change in x is positive 2

Therfore, slope is [tex]\frac{2}{4}[/tex]

or, 1/2

Answer:

This is the equation we have to use in order to find the eigenvalues and eigenvectors of matrix A:

det(A - λI) = 0

where λ is the eigenvalue and I is the identity matrix.

So we have:

A - λI = [ -5-λ 1 2 ]

[ 0 -2-λ 0 ]

[ 4 2 -3-λ ]

And the determinant is:

det(A - λI) = (-5-λ)(-2-λ)(-3-λ) - 8(2+λ) = -λ^3 + 4λ^2 + 23λ - 40

We can factor this to get:

det(A - λI) = -(λ - 5)(λ - 2)(λ + 4)

So the eigenvalues are λ1 = 5, λ2 = 2, and λ3 = -4.

To find the eigenvectors for each eigenvalue, we need to solve the system of linear equations:

(A - λI)x = 0

For λ1 = 5, we have:

[ -5-5 1 2 ] [ x1 ] [ 0 ]

[ 0 -2-5 0 ] [ x2 ] = [ 0 ]

[ 4 2 -3-5] [ x3 ] [ 0 ]

which simplifies to:

[ -10 1 2 ] [ x1 ] [ 0 ]

[ 0 -7 0 ] [ x2 ] = [ 0 ]

[ 4 2 -8 ] [ x3 ] [ 0 ]

We can solve this system to get:

x1 = -1/2 x2

x3 = -1/2 x2

So the eigenvector for λ1 = 5 is:

v1 = [ x1, x2, x3 ] = [ -1/2, 1, -1/2 ]

For λ2 = 2, we have:

[ -5-2 1 2 ] [ x1 ] [ 0 ]

[ 0 -2-2 0 ] [ x2 ] = [ 0 ]

[ 4 2 -3-2] [ x3 ] [ 0 ]

which simplifies to:

[ -7 1 2 ] [ x1 ] [ 0 ]

[ 0 -4 0 ] [ x2 ] = [ 0 ]

[ 4 2 -5 ] [ x3 ] [ 0 ]

We can solve this system to get:

x1 = -2/3 x2

x3 = -4/3 x2

So the eigenvector for λ2 = 2 is:

v2 = [ x1, x2, x3 ] = [ -2/3, 1, -4/3 ]

For λ3 = -4, we have:

[ -5+4 1 2 ] [ x1 ] [ 0 ]

[ 0 -2+4 0 ] [ x2 ] = [ 0 ]

[ 4 2 -3+4] [ x3 ] [ 0 ]

which simplifies to:

[ -1 1 2 ] [ x1 ] [ 0 ]

[ 0 2 0 ] [ x2 ] = [ 0 ]

[ 4 2 1 ] [ x3 ] [ 0 ]

We can solve this system to get:

x1 = -2/5 x2

x3 = 1/2 x2

So the eigenvector for λ3 = -4 is:

v3 = [ x1, x2, x3 ] = [ -2/5, 0, 1/2 ]

To determine matrix A's sign definiteness, we must compute the determinant of all of A's primary submatrices. A square submatrix generated by eliminating the equal amount of rows and columns from the top left corner of A is known as a primary submatrix.

The principal submatrices of A are:

A1 = [-5]

A2 = [ -5 1 ]

[ 0 -2 ]

A3 = [ -5 1 2 ]

[ 0 -2 0 ]

[ 4 2 -3 ]

The determinants of these matrices are:

det(A1) = -5

det(A2) = (-5)(-2) - (0)(1) = 10

det(A3) = (-5)(-2)(-3) + 2(0)(4) + (1)(0)(2) - (-3)(1)(4) - 2(0)(-5) - (-2)(2)(-5) = -92

Because A1's determinant is negative, A is not positive definite. A is not negative definite since the determinants of A1 and A2 have different signs. As a result, A is infinite.

The volume of the leaning regular hexagonal prism, can be found to be 351. 89 inch ³.

A right triangle can be constructed in a regular hexagon by linking the midpoint of one side and a vertex to the center. This specially-crafted triangle comprises two legs, one of which is half the size of the primary side (acting as the hypotenuse) while the remaining leg lies parallel to the hexagon's apothem (height).

The height of the prism can be found with cosine to be:

h = slanted height x Cos (angle )

h = 11 x Cos ( 70 degrees )

h = 3. 762 inches

We can then find the volume of the leaning hexagonal prism to be:

= Area x Height

= 93. 528 x 3. 762

= 351. 89 inch ³

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Answer: 8.625

Step-by-step explanation:The eight stays an 8 snd 5:8 is .625. Add these together and you get 8.625.

Based on calculation, the highest hourly wage is $2,840 per month which gives us $17.75 per hour.

For purpose of making a comparison, we can calculate the hourly wage for each option.

Hourly wage for $34,000 per year:

= $34,000 / (40 hours per week * 52 weeks per year)

= 16.3461538

= $16.35 per hour.

Hourly wage for $2,840 per month:

= $2,840 / (4 weeks per month * 40 hours per week)

= $17.75 per hour

Hourly wage for $650 per week:

= $650 / 40 hours per week

= $16.25 per hour

Hourly wage for $18 per hour.

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The building is, 246 feet tall.

Since, We know that;

Two or more triangles will always be referred to as similar if on comparing their corresponding properties, some common relations holds. Some of the relations are relations between their length of sides, relations among their internal angles etc.

To determine the height of the building, we have;

height of pole = 7 ft

total length of the building's shadow = 167 ft

the length of the shadow of the pole = 4.75 ft

Let the height of the building be represented by h. On comparison, we have;

4.75/ 167 = 7/h

4.75h = 167 x 7

        = 1169

h = 1169/ 4.75

h  = 246.11

Hence, The building is 246 ft. tall.

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The calculated value of the perimeter of the scaled rectangle is 120 inches

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

Perimeter =  48 in

Scale factor = 2.5

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

Perimeter of the scaled rectangle = Perimeter * Scale factor

Substitute the known values in the above equation

Perimeter of the scaled rectangle = 48 * 2.5

Evaluate

Perimeter of the scaled rectangle = 120

Hence, the perimeter of the scaled rectangle is 120

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[tex]PEk = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex] which is the desired equation.

An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Let P(ej) denote the probability of outcome ej for j=1,2,...,N. Then we know that:

P(ej+1) = 2P(ej) for j=1,2,...,N-1

Also, we know that the sum of probabilities of all outcomes is 1:

P(e1) + P(e2) + ... + P(eN) = 1

We can use the geometric sequence formula to find P(ek) in terms of P(e1) as follows:

[tex]P(ek) = P(e1) * (2^{(k-1)})[/tex]

Substituting for P(ej+1) in terms of P(ej) gives:

[tex]P(ej+1) = 2P(ej) = 2^1 * P(ej)[/tex]

[tex]P(ej+2) = 2P(ej+1) = 2^2 * P(ej)[/tex]

...

[tex]P(ek) = 2^{(k-1)} * P(e1)[/tex]

We can sum the probabilities of all outcomes to get:

[tex]1 = P(e1) + 2P(e1) + 2^2 P(e1) + ... + 2^{(N-1)}P(e1)[/tex]

Using the formula for the sum of a geometric series, we get:

[tex]1 = P(e1) * [(2^N - 1)/(2 - 1)][/tex]

Simplifying, we get:

[tex]P(e1) = 1/(2^N - 1)[/tex]

Substituting this value for P(e1) in the expression for P(ek) gives:

[tex]P(ek) = (1/(2^N - 1)) * (2^{(k-1)})[/tex]

Simplifying, we get:

[tex]P(ek) = 2^{(k-N)} / (2 - 1/2^{(N-k+1)})[/tex]

Multiplying by 2/2, we get:

[tex]P(ek) = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex]

Substituting k=N, we get:

P(Ek) = 1/2

Substituting k=1, we get:

[tex]P(E1) = 2^{(N-1)} / (2^N - 1)[/tex]

Therefore, we have shown that:

[tex]P(Ek) = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex]

for k=1,2,...,N.

To show that PEk=2k-1/2N-1, we can substitute k=N-j+1 for j=1,2,...,N:

[tex]PEk = P(EN-j+1) = 2^{(N-j)} / (2^j - 1)[/tex]

Letting t = j-1, we can rewrite this as:

[tex]PEk = 2^{(N-t-1)} / (2^{(t+1)} - 1[/tex]

Multiplying the numerator and denominator by 2, we get:

[tex]PEk = 2^{(N-t)} / (2^{(t+2)} - 2)[/tex]

Substituting t = N-k, we get:

[tex]PEk = 2^{(k-1)} / (2^{(N-k+1)} - 1)[/tex]

which is the desired result.

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The calculated median and the mean gasoline prices per gallon in California are 3.25 and 4.49, respectively

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

Prices = 3.14, 3.21, 3.28, 3.53

Start by sorting the number of prices in ascending order

So, we have

3.14, 3.21, 3.28, 3.53

As a general rule.

The median is the middle number

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

Median = middle number = 1/2 *(3.21 + 3.28)

Evaluate

Median = middle number = 3.25

For the mean, we have

Mean = (3.14 + 3.21 + 3.28 + 3.53)/3

Evaluate

Mean = 4.49

Hence, the value of the median is 3.25

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Answer:

y=3x

Step-by-step explanation:

y=kx

21/7=3

15/5=3

6/2=3

Each relationship in the table has a constant of proportionality of 3.

Answer:

1984

________________________________________________________

Given:

To find the inverse of f(x), we first switch x and y and then solve for y. So, x = y-3/y+4, which we can rewrite as x(y+4) = y-3. Simplifying, we get xy + 4x = y-3, and then we can isolate y on one side: y-xy = 4x-3. Factoring out y on the left side, we get y(1-x) = 4x-3, and then we can divide both sides by (1-x) to get y = (4x-3)/(1-x). This is our inverse function.

Find:

To find the inverse of g(x), we follow the same process of switching x and y and solving for y. So, x = 4y+3/1-y, which we can rewrite as x(1-y) = 4y+3. Simplifying, we get -xy + y = 4x+3, and then we can isolate y on one side: y(-x+1) = 4x+3. Dividing both sides by (-x+1), we get y = (4x+3)/(-x+1). This is our inverse function.

Solve:

As for the given set of values, we have 187, 191, 202, 209, 218, and 1984. The outlier is obviously 1984, and its presence will not affect the range because the range is simply the difference between the largest and smallest values, which will be the same regardless of the presence of an outlier. However, the outlier will greatly affect the interquartile range, which is the difference between the upper and lower quartiles. This is because the upper and lower quartiles are the median of the upper half and lower half of the data, respectively, and including an outlier in one of these halves can greatly skew the median and thus the interquartile range.

The quadrilateral is inscribed inside the circle and x = 44

Given data ,

Let the quadrilateral be inscribed inside the circle as shown in the figure

Now , the measure of angles are

m∠CFE = (2x + 6)°

m∠CDE = (2x − 2)°

And , opposite sides of the angles of quadrilateral are supplementary

So , m∠CFE + m∠CDE = 180°

On simplifying , we get

2x + 6 + 2x - 2 = 180

4x + 4 = 180

4x = 176

x = 44

Hence , the angle is solved and x = 44

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The complete question is attached below :

Quadrilateral CDEF is inscribed in circle A.

If m∠CFE = (2x + 6)° and m∠CDE = (2x − 2)°, what is the value of x?

A) 22

B) 44

C) 46

D) 89

Answer: 44

Step-by-step explanation:

i took the test

Answer:

$97.50

Step-by-step explanation:

Purchase Price:

$150

Discount:

(150 x 35)/100 = $52.50

Final Price:

150 - 52.50 = $97.50

Check the picture below.

the slope goes by several names

• average rate of change

• rate of change

• deltaY over deltaX

• Δy over Δx

• rise over run

• gradient

• constant of proportionality

however, is the same cat wearing different costumes.

[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x_1=0\\ x_2=15 \end{cases}\implies \cfrac{f(15)-f(0)}{15 - 0}\implies \cfrac{11-5}{15}\implies \cfrac{6}{15}\implies \cfrac{2}{5}[/tex]

Answer: 3 meters

Step-by-step explanation:

The area of the rectangular region is given by:

(18 + 2x) * (30 + 2x) = 18*30 + 36x + 60x + 4x^2

Simplifying this expression, we get:

4x^2 + 96x - 1300 = 0

Dividing both sides by 4, we get:

x^2 + 24x - 325 = 0

Now we can solve for x using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = 24, and c = -325.

Plugging in these values, we get:

x = (-24 ± sqrt(24^2 - 4(1)(-325))) / 2(1)

x = (-24 ± sqrt(936)) / 2

x = (-24 ± 30) / 2

x = 3 or -27

Since the width of the path cannot be negative, the only possible solution is x = 3 meters. Therefore, the width of the path surrounding the pool is 3 meters.

The end behavior of the given function is As x → –∞, f(x) → –∞, and as x → ∞, f(x) → –∞.

The term "end behavior" describes the behaviour or trend of a function's graph as its x-values go closer to positive or negative infinity. It explains how the function "ends" or acts when x is extremely large (positive infinity) or extremely small (negative infinity). The degree and leading coefficient of the polynomial formulation of the function can be used to predict the final behaviour. For instance, the left and right ends of the graph will both point upward for a function with an odd-degree polynomial and a positive leading coefficient (towards positive infinity).

For the function f(x) = -2∛x as x approaches negative infinity because the function has a negative coefficient in front of the cube root, which means that as x decreases significantly, the absolute value of the cube root increases. The function also approaches negative infinity as x approaches very big (positive) values and the cube root approaches very small absolute values. As a result, the function's final behaviour is that it approaches negative infinity as x gets closer to both negative and positive infinity.

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The end behavior of the given function is As x → –∞, f(x) → –∞, and as x → ∞, f(x) → –∞.

The term "end behavior" describes the behaviour or trend of a function's graph as its x-values go closer to positive or negative infinity. It explains how the function "ends" or acts when x is extremely large (positive infinity) or extremely small (negative infinity). The degree and leading coefficient of the polynomial formulation of the function can be used to predict the final behaviour. For instance, the left and right ends of the graph will both point upward for a function with an odd-degree polynomial and a positive leading coefficient (towards positive infinity).

For the function f(x) = -2∛x as x approaches negative infinity because the function has a negative coefficient in front of the cube root, which means that as x decreases significantly, the absolute value of the cube root increases. The function also approaches negative infinity as x approaches very big (positive) values and the cube root approaches very small absolute values. As a result, the function's final behaviour is that it approaches negative infinity as x gets closer to both negative and positive infinity.

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Answer:

The function f(x) = -2∛x represents a cube root function that is multiplied by -2.

As x approaches negative infinity, the cube root of x becomes more negative, and when it is multiplied by -2, it becomes more positive. Therefore, as x → –∞, f(x) → ∞.

As x approaches positive infinity, the cube root of x becomes more positive, and when it is multiplied by -2, it becomes more negative. Therefore, as x → ∞, f(x) → –∞.

Therefore, the correct answer is: As x → –∞, f(x) → ∞, and as x → ∞, f(x) → –∞. D.

Answer:

The function f(x) = -2∛x represents a cube root function that is multiplied by -2.

As x approaches negative infinity, the cube root of x becomes more negative, and when it is multiplied by -2, it becomes more positive. Therefore, as x → –∞, f(x) → ∞.

As x approaches positive infinity, the cube root of x becomes more positive, and when it is multiplied by -2, it becomes more negative. Therefore, as x → ∞, f(x) → –∞.

Therefore, the correct answer is: As x → –∞, f(x) → ∞, and as x → ∞, f(x) → –∞. D.

Answer:

No solution

Step-by-step explanation:

-2(11-12x) = -4(1-6x)

Distribute the -2 and -4:

-22 + 24x = -4 + 24x

Subtract 24x from both sides:

-22 = -4

Since this is a contradiction, there is no solution. The equation is inconsistent.

Answer:

No Solution.

Explanation:

When solving for x, the two values do not equate to each other.

Answer:

A data set of the least purchased menu items at a bakery is best shown as a (Report)

Step-by-step explanation:

Have A Good Day :)

(i)  distributed in [tex]4 \times 3 = 12[/tex] different ways. (ii)statistical distribution of the balls is done in one of three ways. [tex](1 + 2 = 3)[/tex]  . (iii) there are 6 ways to divide the balls. [tex](4 + 2 \times 1).[/tex]  (iv)  to distribute the balls   [tex](1+2+1)[/tex].

(i) Both balls and boxes are labeled:

Since one box already contains a ball, there are four options for where to place the first ball, and only three options remain for the second ball. The two balls can be distributed in  [tex]4 \times 3 = 12[/tex] different ways.

(ii) Balls are labeled, but boxes are unlabeled:

There is just one method to arrange the balls if there is only one ball in each box. There are two ways to determine which box receives both balls if only one box possesses them. The statistical distribution of the balls is done in one of three ways.

(1 + 2 = 3).

(iii) Balls are unlabeled, but boxes are labeled:

There are four alternatives available to the box if they are both in the same box. If they are in different boxes, the first ball's box has two options, leaving the second ball with just one. Accordingly, there are 6 ways to divide the balls.

(4 + 2 * 1).

(iv) Both balls and boxes are unlabeled:

There is only one method to arrange the balls if there is one ball in each box. There are two methods to decide which box gets both balls if only one box has both balls.

There is just one method to arrange the balls if neither box incorporates a ball. There are therefore a total of 4 options to distribute the balls (1+2+1).

Therefore,(i)  distributed in [tex]4 \times 3 = 12[/tex] different ways. (ii)statistical distribution of the balls is done in one of three ways. [tex](1 + 2 = 3)[/tex]  . (iii) there are 6 ways to divide the balls. [tex](4 + 2 \times 1).[/tex]  (iv)  to distribute the balls   [tex](1+2+1)[/tex].

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