We start by subdividing [0,5] into n equal width subintervals [x0,x1],[x1,x2],…,[xn−1,xn] each of width Δx. Express the width of each subinterval Δx in terms of the number of subintervals n.

Answer:

The margin of error of the poll is 8%.

Step-by-step explanation:

This is a confidence interval. A confidence interval has both a lower end and an upper end.

The true proportion is the midpoint between the two ends.

The margin of error is the absolute difference between the proportion and the ends(which is the same, upper end - proportion = proportion - lower end),

In this problem, we have that:

The lower end is 52%.

The upper end is 68%.

The proportion is (52 + 68)/2 = 60%.

The margin of error is 60 - 52 = 68 - 60 = 8%.

Answer:

See the proof below.

Step-by-step explanation:

We can define a basis of V with the following elements:

[tex]X_1=\begin{matrix}1 & 0 \\0 & 0 \end{matrix} [/tex]

[tex]X_2=\begin{matrix}0 & 1 \\0 & 0 \end{matrix} [/tex]

[tex]X_3=\begin{matrix}0 & 0 \\1 & 0 \end{matrix} [/tex]

[tex]X_4=\begin{matrix}0 & 0 \\0 & 1 \end{matrix} [/tex]

So then if we define the basis X as following:

[tex] X = [X_1, X_2, X_3, X_4][/tex]

[tex]X =[\begin{pmatrix}1 & 0\\0 & 0\end{pmatrix},\begin{pmatrix}0 & 1\\0 & 0 \end{pmatrix},\begin{pmatrix}0 & 0\\1 & 0\end{pmatrix},\begin{pmatrix}0 & 0\\0 & 1 \end{pmatrix}[/tex]

We see the the dimension for X is 4 [tex] dim (V) = 4[/tex] since the basis have a dimension of 4 [tex] dim (X) =4[/tex]

Final answer:

The vector space V of all 2 x 2 matrices over a field F has a basis consisting of four matrices which are linearly independent and span V. This basis demonstrates that V has a dimension of 4.

Explanation:

In order to prove that the vector space V of all 2 x 2 matrices over a field F has dimension 4, we need to exhibit a basis for V that consists of four linearly independent elements, which also span V. Consider the following 2 x 2 matrices as the candidate basis elements:

These matrices are linearly independent and span the vector space of all 2 x 2 matrices. To show linear independence, assume that a linear combination of these matrices equals the zero matrix:

a ⬑ 1 0 ⬑ + b ⬑ 0 1 ⬑ + c ⬑ 0 0 ⬑ + d ⬑ 0 0 ⬑
⬑ 0 0 ⬑   ⬑ 0 0 ⬑   ⬑ 1 0 ⬑   ⬑ 0 1 ⬑
= ⬑ 0 0 ⬑
⬑ 0 0 ⬑

This equation leads to a = b = c = d = 0, which verifies the linear independence. Since we can represent any 2 x 2 matrix as a linear combination of these four basis matrices, they also span V, fulfilling both criteria for a basis. Hence, there are four basis elements, and therefore, the dimension of V is 4.

Answer:

The slope of the regression line is 0.01.

Step-by-step explanation:

The given regression equation for this bird is

[tex]DD=2.64+0.01D[/tex]        .... (1)

where, DD is dive duration measured in minutes, and D is depth in meters.  

The slope intercept form of a line is

[tex]y=mx+b[/tex]            .... (2)

where, m is slope and b is y-intercept.

On comparing equation (1) and (2), we get

[tex]y=DD,x=D,m=0.01,b=2.64[/tex]

Since, m=0.01, therefore the slope of the regression line is 0.01.

Answer:

y = 0.0765x

Step-by-step explanation:

We have that:

y is the total taxes paid.

x is the total wages earned.

The total taxes paid is a function of the total wages earned.

For wages less than the maximum taxable wage base, Social Security contributions (including those for Medicare) by employees are 7.65% of the employee's wages.

So 7.65% of the total wages earned are paid in taxes. We write the percentage as a decimal, so 7.65/100 = 0.0765

So the answer for a) is:

y = 0.0765x

The relationship between the wages earned (x) and the Social Security taxes paid (y) can be expressed as a linear equation. The equation is y = 0.0765x, which means for every dollar earned, 7.65 cents are paid towards Social Security taxes.

The relationship between the wages earned (x) and the Social Security taxes paid (y) can be expressed as a linear equation. Since the contribution is 7.65% of the wages, the equation becomes y = 0.0765x, where x stands for the wages earned, and y represents the employee's Social Security taxes paid.

This equation means that for every dollar earned, 7.65 cents are paid towards Social Security taxes. So, if an employee earns $1000, you would substitute x with 1000 to find the taxes payable. That is y = 0.0765 * 1000, which equates to $76.50

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Answer:

There is a 40.96% probability that he answers exactly 1 question correctly in the last 4 questions.

Step-by-step explanation:

For each question, there are only two possible outcomes. Either it is correct, or it is not. This means that we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this problem we have that:

There are four questions, so n = 4.

Each question has 5 options, one of which is correct. So [tex]p = \frac{1}{5} = 0.2[/tex]

What is the probability that he answers exactly 1 question correctly in the last 4 questions?

This is [tex]P(X = 1)[/tex]

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 1) = C_{4,1}*(0.2)^{1}*(0.8)^{3} = 0.4096[/tex]

There is a 40.96% probability that he answers exactly 1 question correctly in the last 4 questions.

Answer:

0.41

Step-by-step explanation:

kahn

Answer:

2(x+6)(x-3)

Step-by-step explanation:

Factor the GCF out of the trinomial on the left side of the equation.

[tex]2x^2 + 6x - 36 =2(x^2 + 3x - 18)[/tex]

Greatest common factor of 2, 6, 18 is 2

so GCF is 2

divide each term when we take out GCF 2

so [tex]2(x^2 + 3x - 18)[/tex]

now factor the trinomial

product is -18 and sum is +3

6 times -3 is -18  and 6-3=3

[tex]2(x^2+3x-18)\\2(x+6)(x-3)[/tex]

If the coefficient matrix has a pivot in each column, it means that it is shaped like this:

[tex]A=\left[\begin{array}{cccc}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}\\0&a_{2,2}&a_{2,3}&a_{2,4}\\0&0&a_{3,3}&a_{3,4}\\0&0&0&a_{4,4}\end{array}\right][/tex]

So, the correspondant system

[tex]Ax = b[/tex]

will look like this:

[tex]\left[\begin{array}{cccc}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}\\0&a_{2,2}&a_{2,3}&a_{2,4}\\0&0&a_{3,3}&a_{3,4}\\0&0&0&a_{4,4}\end{array}\right]\cdot \left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right] = \left[\begin{array}{c}b_1\\b_2\\b_3\\b_4\end{array}\right][/tex]

This turn into the following system of equations:

[tex]\begin{cases}a_{1,1}x_1+a_{1,2}x_2+a_{1,3}x_3+a_{1,4}x_4=b_1\\a_{2,2}x_2+a_{2,3}x_3+a_{2,4}x_4=b_2\\a_{3,3}x_3+a_{3,4}x_4=b_3\\a_{4,4}x_4=b_4\end{cases}[/tex]

The last equation is solvable for [tex]x_4[/tex]: we easily have

[tex]x_4=\dfrac{b_4}{a_{4,4}}[/tex]

Once the value for [tex]x_4[/tex] is known, we can solve the third equation for [tex]x_3[/tex]:

[tex]x_3 = \dfrac{b_3-a_{3,4}x_4}{a_{3,3}}[/tex]

(recall that [tex]x_4[/tex] is now known)

The pattern should be clear: you can use the last equation to solve for [tex]x_4[/tex]. Once it is known, the third equation involves the only variable [tex]x_3[/tex]. Once

Answer:

a) a binomial distribution with mean 50

Step-by-step explanation:

Given that an  airplane has a front nad a rear door that are bother opened to allow passengers to exit when the plane lands. the plane has 100 passengers.

These 100 passengers can select either back door or front door with equal probability (assuming)

so probability for selecting front door = 0.5

No of passengers =100

Each passenger is independent of the other

Hence X no of passengers exiting through the front door is binomial with

p =0.5 and n =100

Mean of the variable X = np = 100(0.5) = 50

Variance of X = 100(0.5)(0.5)

Hence std dev = 10(0.5) = 5

So correct answers are

a) a binomial distribution with mean 50

Answer:  (16.9914, 28.2286).

Step-by-step explanation:

The formula to find the confidence interval for population mean is given by :-

[tex]\overline{x}\pm z^*\dfrac{\sigma}{\sqrt{n}}[/tex]

, where [tex]\overline{x}[/tex] = Sample mean

[tex]\sigma[/tex]= Population standard deviation

n= Sample size.

z* = Critical value.

Let μ be the mean change in score  in the population of all high school seniors.

As per given ,  we have

n= 350

[tex]\overline{x}=22.61[/tex]

[tex]\sigma=53.63[/tex]

The critical z-value for 95% confidence interval is z*= 1.96 [From z-table]

Substitute all the value in formula , we get

[tex]22.61\pm (1.96)\dfrac{53.63}{\sqrt{350}}[/tex]

[tex]=22.61\pm (1.96)\dfrac{53.63}{18.708287}[/tex]

[tex]=22.61\pm (1.96)(2.8666)[/tex]

[tex]=22.61\pm (5.6186)[/tex]

[tex]=(22.61-5.6186,\ 22.61+5.6186) =(16.9914,\ 28.2286)[/tex]

Hence, the 95% confidence interval for [tex]\mu[/tex] is (16.9914, 28.2286).

Answer:

0.5 is the probability that a randomly selected home will have a Jacuzzi given that it has a swimming pool.              

Step-by-step explanation:

We are given the following in the question:

S: Homes in Miami have a swimming pool

J: Homes in Miami have a jacuzzi

[tex]P(S) = 60\% = 0.6\\P(S\cap J) = 30\% = 0.3[/tex]

We have to find the probability that a randomly selected home will have a Jacuzzi given that it has a swimming pool.

Thus, we have to calculation the conditional probability of having a jacuzzi given the house has a swimming pool.

[tex]P(J|S) = \dfrac{P(J\cap S)}{P(S)}\\\\P(J|S) = \displaystyle\frac{0.3}{0.6} = 0.5[/tex]

0.5 is the probability that a randomly selected home will have a Jacuzzi given that it has a swimming pool.

Answer:

[tex] [tex] lim_{x \to 8} (1+3\sqrt{x})(1-6x^2 +x^3)[/tex]=[tex]1-384 +512+3\sqrt{8} -18(8)^{5/2} +3 (8)^{7/2} =1223.601[/tex]

And the limit on this case exists.

Step-by-step explanation:

We want to find the following limit:

[tex] lim_{x \to 8} (1+3\sqrt{x})(1-6x^2 +x^3)[/tex]

First we can distribute the polynomials like this:

[tex] lim_{x \to 8} (1-6x^2 +x^3+3\sqrt{x} -18 x^{5/2} +3x^{7/2})[/tex]

And Now we can use the distributive property for the limit and we got:

[tex] lim_{x \to 8} 1 - 6 lim_{x \to 8} x^2 + lim_{x \to 8} x^3 +3 lim_{x \to 8} \sqrt{x} -18 lim_{x \to 8} x^{5/2} + 3 lim_{x \to 8} x^{7/2}[/tex]

And now we can evaluate the limit and we got:

[tex] [tex] lim_{x \to 8} (1+3\sqrt{x})(1-6x^2 +x^3)[/tex]=[tex]1-384 +512+3\sqrt{8} -18(8)^{5/2} +3 (8)^{7/2} =1223.601[/tex]

And the limit on this case exists.

To solve limit problems in mathematics, limit laws are often very useful. In this specific case, as the function is a polynomial and defined for all real number values, a direct substitution of x=8 into the function is sufficient. Therefore, the limit as x approaches 8 for function 1 + 3x5 - 6x2 + x3 is calculable.

In the field of mathematics, limit laws are used quite frequently for evaluating limits. In this case, we want to calculate the limit as x approaches 8 for the function 1 + 3x5 - 6x2 + x3.

For a given polynomial function like this one, an easy and very straightforward approach is to substitute the value x is approaching (in this scenario, x = 8) directly into the polynomial function.

So, after substitution, our function becomes: 1 + 3*(8)^5 - 6*(8)^2 + (8)^3. Simplifying it further, the limit as x approaches 8 of this function gives us a definite numeric value.

Always remember while applying limit laws, you might at times need the limit laws to evaluate complex limit problems but in this given scenario, direct substitution works perfectly fine because this polynomial function is defined for all real number values of X.

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Answer:

3.2 pi

Step-by-step explanation:

Given 3 curves are:

y = square root ( x / 2)

y = 0

x = 2

Use washer method for hollow volumes.

Step 1: Compute A (y)

A ( y ) = pi * ( f_1 (y) ^2 - f_2 (y) ^2)

where,

f_1 (y) is the function further away from y axis

f_2 (y) is the function closer to y axis

f_1 (y) = 2

f_2 (y) = 2*y^2

A ( y ) = pi * ( 2 ^2 - (2*y) ^2)

A (y) = pi * (4 - 4*y^2)

A (y) = 4*pi * (1 - y^2)

Step 2: Compute V (y)

[tex]V = \int\limits^1_0 {A (y)} \, dy \\V = 4*pi\int\limits^1_0 {1 - y^2} \, dy\\\\V = 4 * pi* (y - 0.2 y^5) \limits^1_0\\\\V = 4*pi*(1 - 0.2)\\\\V = 3.2 pi[/tex]

Answer: V = 3.2 pi

Answer:

[tex]x + 3y -z - 3 = 0[/tex]      

Step-by-step explanation:

We have to find the equation of plane that is parallel to the vectors

[tex]\langle 3,0,3\rangle, \langle0,1,3\rangle[/tex]

The plane also passes through the point (2,0,-1).

Hence, the equation of plane s given by:

[tex]\displaystyle\left[\begin{array}{ccc}x-2&y-0&z+1\\3&0&3\\0&1&3\end{array}\right]\\\\=(x-2)(0-3) - (y-0)(9-0) + (z+1)(3-0)\\=-3(x-2)-9y+3(z+1)\\\Rightarrow -3x + 6 - 9y + 3z + 3 = 0\\\Rightarrow 3x + 9y -3z -9 = 0\\\Rightarrow x + 3y -z - 3 = 0[/tex]

It is the required equation of plane.

Answer:

Central angle= 1.15 radians

Step-by-step explanation:

[tex]Arc\,\,length=s= 1.22\,m\\Radius=r=1.06\,m\\\\Central\,\, angle=\theta=?\\\\Using\\\\ s=r\theta\\\\\theta=\frac{s}{r}\\\\\theta= \frac{1.22}{1.06}\\\\\theta=1.15 \,rad[/tex]

Answer:

I'm pretty sure it would be 345, just add the two 178 and 167

Answer:its A

Step-by-step explanation:it was

Answer:

all work is shown and pictured

Answer:The total cost of the Sony Playstation is $179.9096

Step-by-step explanation:

The initial or regular cost of the Sony Playstation is $172.99.

The tax rate is 4%. Therefore, the value of the sales tax would be

4/100 × 172.99 = 0.04 × 172.99 = $6.9196

The total cost of the Sony Playstation would be the sum of the regular price and the sales tax. It becomes

172.99 + 6.9196 = $179.9096

Answer:

The answer is B

Step-by-step explanation:

Answer:

B. The second graph

Step-by-step explanation:

edge 2021 math assignment

Answer:

For x=45

sample proportion=45/140=0.321

z=(0.321-0.30)/sqrt(0.3*(1-0.3)/140)

z=0.54

For x=47

sample proportion=47/140=0.336

z=(0.336-0.30)/sqrt(0.3*(1-0.3)/140)

z=0.93

Now,

P(0.54<z<0.93)=P(z<0.93)-P(z<0.54)

=0.8238-0.7054

=0.118

So,correct option is 0.119

Answer: Area of triangle is √3 / 2

Step-by-step explanation:

The explanation can be found in the attached in picture

Answer: a. 0.40   b. 0.23  c . 0.435   d . 0.25

Step-by-step explanation:

                                   melanin      content    Total

                                            high   low

moisture   high                     13      10                23

content    low                       47      30                77

 Total                                   60      40               100

Let A denote the event that a sample has low melanin content, and let B denote the event that a sample has high moisture content.

a) Total skin samples has low melanin content = 10+30=40

P(A)=[tex]\dfrac{40}{100}=0.40[/tex]

b) Total skin samples has high moisture content = 13+10=23

P(B) =[tex]\dfrac{23}{100}=0.23[/tex]

c) A ∩ B =  Total skin samples has both low melanin content and high moisture content =10

P(A ∩ B) =[tex]\dfrac{10}{100}=0.10[/tex]

Using conditional probability formula , [tex]P (A|B)=\dfrac{P(A\cap B)}{P(B)}[/tex]

[tex]P (A|B)=\dfrac{0.10}{0.23}=0.434782608696\approx0.435[/tex]

d)  [tex]P (B|A)=\dfrac{P(A\cap B)}{P(A)}[/tex]

[tex]P (B|A)=\dfrac{0.10}{0.40}=0.25[/tex]

Answer:

a) No it doesn't mean that linear model is inappropriate

b) No. The prediction using this model will not be accurate.

Step-by-step explanation:

a)

For answering this part, firstly consider the concept of [tex]R^{2}[/tex]

The [tex]R^{2}[/tex] also known as coefficient of determination is used to determine the amount of variability in dependent variable is explained by the linear model. Lower [tex]R^{2}[/tex] depicts that less variation of dependent is explained by the independent variable using the linear model. The linearity of model is determined by scatter plot. Thus, if the [tex]R^{2}[/tex] is lower, it doesn't mean that linear model is inappropriate.

b)

The predictions made by the model having lower [tex]R^{2}[/tex] value are erroneous. The model is used for prediction if the linear model explains the larger portion of variability in dependent variation. If the predictions made from the model that have lower [tex]R^{2}[/tex] value then the predicted values will not be close to the actual value and thus residuals will not be minimum as residuals are the difference of actual and predicted values.

The result is [tex]\frac{4}{7}[/tex]

Step-by-step explanation:

In this problem, we are asked to find how many 7/8 cup servings are in 1/2 of a cup of juice.

Mathematically, this is equivalent to divide 1/2 by 7/8. So we can write:

[tex]\frac{1/2}{7/8}[/tex]

This can be rewritten as a multiplication by reversing the denominator:

[tex]\frac{1}{2}\cdot \frac{8}{7}[/tex]

Now we can perform the multiplication of both the numerator and the denominator:

[tex]\frac{1\cdot 8}{2\cdot 7}=\frac{8}{14}[/tex]

And simplifying (dividing by 2),

[tex]\frac{8}{14}=\frac{4}{7}[/tex]

Learn more about  fractions:

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There are 4/7 servings of 7/8 cup in 1/2 cup of juice.

To determine the number of 7/8 cup servings in 1/2 of a cup of juice, divide the 1/2 cup of juice by 7/8 cup.

Now, the reciprocal of 7/8 and multiplying it by 1/2.

Reciprocal of 7/8 = 8/7

Now, perform the multiplication:

= (1/2 cup) * (8/7)

= (1 * 8) / (2 * 7)

= 8/14

= 4/7

Therefore,4/7 servings of 7/8 cup in 1/2 cup of juice.

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Answer:

5

Step-by-step explanation:

Assuming both players can answer the same question, the minimum number of questions is the smallest number that when multiplied by either 0.60 or 0.80 yields a whole number.

Let x be the number of questions, solving by trial and error:

[tex]if\ x=2\\x*0.8=1.6\\x*0.6=1.2\\\\if\ x=3\\x*0.8=2.4\\x*0.6=1.8\\\\if\ x=4\\x*0.8=3.2\\x*0.6=2.4\\\\if\ x=5\\x*0.8=4\\x*0.6=3\\\\[/tex]

Therefore, the minimum number of questions in the game is 5.

To find the minimum number of questions in a game where one person answers 80% correctly and another answers 60% correctly, calculate the LCM of the fractions' denominators. The result is 5 questions.

You and your friend have different accuracy rates when answering questions in a game. You answer 80% of the questions correctly, while your friend answers 60% of the questions correctly. To find the minimum number of questions in the game, we need to ensure that both percentages can correspond to whole numbers of questions.

In a game with 5 questions:

Therefore, the minimum number of questions in this game is 5.

Answer:

D) 0.0902

Step-by-step explanation:

Data provided in the question:

Probability of burglary, p = [tex]\frac{661}{400,000}[/tex]

= 0.00165      

q = 1 - p                

or                    

q = 1 - 0.00165                      

or                    

q = 0.99835                              

Now,

P(r ≥ 2) = 1 - P(r < 2)

= 1 - [ P(0) + P(1) ]

= 1 - [  [tex]^{317}C_0(0.00165)^0(0.99835)^{317-0}+^{317}C_1(0.00165)^1(0.99835)^{317-1}[/tex] ]

[ as P(x) = [tex]^nC_rp^rq^{n-r}[/tex]]

= 1 - [ 0.593 + 0.3168]

= 1 - 0.9098

= 0.0902

Hence,

Option (D) 0.0902

Answer:

Step-by-step explanation:

Since quality points to letter grades are A=​4; B=​3; C=​2; D=​1; F=0, weighted mean is the sum of the qulity points times corresponding credit hours divided by the total credit hours:

[tex]\frac{(4*4) + (1*2) + (4*2) + (2*3) + (3*1)}{12}[/tex] ≈ 2.92

Since 2.92<3.0, the student is not in Dean's list.

Answer: You have only provided one Differential Equation (DE), it looks like you intended listing more.

The equation you wrote contains an incorrect d²x/dt, it is likely to be d²x/dt² + 5dx/dt + 10x = 0, which is linear. Unless it is (dx/dt)² + 5dx/dt + 10x = 0, then it is nonlinear.

Not to worry though, I will explain what linear and nonlinear DE's are.

Step-by-step explanation:

LINEAR DE: This is the kind of DE in which the functions of the dependent variable are linear. There are no powers of the dependent variable and/or its derivatives, there are no products of the dependent variable and its derivative, there are no functions of the dependent variable like cos, sin, exp, etc.

Example:

* 5d²x/dt² + dx/dt - x = 2t

This is linear, as it satisfies all the conditions.

NONLINEAR DE: If any condition explained for linear DE is not satisfied, then it is called nonlinear.

Example:

* d²x/dt² - sinx = 0

This is nonlinear because of the presence of sinx.

* d²x/dt² + xdx/dt = 0

This is nonlinear because of the product of the dependent variable, x, and its derivative, dx/dt.

* d²x/dt² + x² = 0

This is nonlinear because a function of the dependent variable is not linear. You shouldn't have x².

* (dx/dt)³ + 3dx/dt = 0 is equally nonlinear. You can't have nonlinear functions of the dependent variable or its derivatives.

I hope this helps answer the remaining parts of your question.

The length of the angle bisector of angle ∠A is approximately 4.62 feet.

To find the length of the angle bisector of angle ∠A in triangle ΔABC, we can use the Angle Bisector Theorem, which states that in a triangle, the angle bisector of a vertex divides the opposite side into segments proportional to the adjacent sides.

In triangle ΔABC, let AD be the angle bisector of ∠A, where D lies on BC. According to the Angle Bisector Theorem:

AC/CD = AB/BD

Given AC = 5 ft and AB = 13 ft, we can plug in these values:

5/CD = 13/BD

To find BD, we use the Pythagorean theorem:

BD = √(AB² - AD²) = √(13² - 5²) = √(169 - 25) = √144 = 12 ft

Now, using the Angle Bisector Theorem:

5/CD = 13/12

Cross-multiply:

5 × 12 = 13 × CD

CD = (5 × 12) / 13 = 60 / 13 ≈ 4.62 ft

Answer:

dont see much information here but as far as lawsuits go id aim for the highest answer

Step-by-step explanation:

_____________________________________

Answer:

a) 1820 ways

b) 43680 ways

Step-by-step explanation:

When the order of the choices is relevant we use the permutation formula:

[tex]P_{n,x}[/tex] is the number of different permutations of x objects from a set of n elements, given by the following formula.

[tex]P_{n,x} = \frac{n!}{(n-x)!}[/tex]

When the order of choices is not relevant we use the combination formula:

[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

In this problem, we have that:

[tex]x = 4, n = 16[/tex]

(a) How many ways can this be done, if the order of the choices is not relevant?

[tex]C_{16,4} = \frac{16!}{4!(12)!} = 1820[/tex]

(b) How many ways can this be done, if the order of the choices is relevant?

[tex]P_{16,4} = \frac{16!}{(12)!} = 43680[/tex]

We can choose 4 objects from 16 in 1820 ways if order doesn't matter (combination), and in 43680 ways if order does matter (permutation).

The subject of this question is combinatorial mathematics. You're being asked to calculate combinations and permutations.

(a) If the order of the choices is not relevant, we are dealing with a combination. The formula for a combination is C(n, r) = n! / [r!(n-r)!], where n is the number of objects and r is the number of objects chosen. In this case, n = 16 and r = 4, so C(16, 4) = 16! / [4!(16-4)!] = 1820 combinations.

(b) If the order of the choices is relevant, we are dealing with a permutation. The formula for a permutation is P(n, r) = n! / (n-r)!. Again, n = 16 and r = 4, so P(16, 4) = 16! / (16-4)! = 43680 permutations.

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