Can someone plz help me solved this problem I need help plz help me! Will mark you as brainiest!

Answer:

(B)[tex]\frac{10}{10}-\frac{7}{10}=\frac{3}{10}[/tex]

Step-by-step explanation:

The question is Incomplete. Find the complete question in the attachment.

Number of muffins made by Franco =12

Number of Badly Burnt Muffins =2

Number of non burned muffins =12-2=10

Franco served [tex]\frac{7}{10} $ of the non burned muffins = $ \frac{7}{10}*10=7$ muffins[/tex]

Therefore, the number of non burned muffins that remains =10-7 =3

We can then say the fraction of the non-burned muffins that remains[tex]=\frac{3}{10}[/tex]

Therefore, the correct equation is in Option B:

[tex]\frac{10}{10}-\frac{7}{10}=\frac{3}{10}[/tex]

Answer:

True

Step-by-step explanation:

Explanation:-

The equation of the standard hyperbola is  

                         [tex]\frac{x^{2} }{a^{2} } - \frac{y^{2} }{b^{2} } =1[/tex]

Answer:

It is True!

Step-by-step explanation:

Answer: $142.5

Step-by-step explanation: If the price is being marked up by 50%, you take 50% or half of the original price of $95, which is 47.5, and add it to your original price.

95 + 47.5 = 142.5

Answer:

$142.5

Step-by-step explanation:

If the price is being marked up by 50%, you take 50% or half of the original price of $95, which is 47.5, and add it to your original price.

95 + 47.5 = 142.5

Answer:

its D on ed

Step-by-step explanation:

3 sqrt x^9 * x

The expression is [tex]x^{10/3}[/tex]

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

∛[tex]x^{10}[/tex]

This can be written as,

= [tex]x^{10/3}[/tex]

Thus,

The expression is [tex]x^{10/3}[/tex]

Learn more about expressions here:

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Answer:

[tex]BD= 4[/tex]

Answer:

BD would equal 4

Step-by-step explanation:

Quadratic formula:

[tex]x=\frac{-b±\sqrt{b^{2}-4ac } }{2a}[/tex]

What do we have:

a=4

b=-10

c=5

Substitude:

[tex]x=\frac{10±\sqrt{b^{2}-4(4)(5)}}{2(a)}[/tex]

Solve:

[tex]x=\frac{10±\sqrt{(-10)^{2}-4(4)(5)}}{2(4)}\\\\x=\frac{10±\sqrt{100-4(20)}}{8}\\\\x=\frac{10±\sqrt{100-4(20)}}{8}\\\\x=\frac{10±\sqrt{100-80}}{8}\\\\x=\frac{10±\sqrt{20}}{8}\\\\x=\frac{10±2\sqrt{5 }}{8}\\\\x=\frac{2(±5\sqrt{5) }}{2(4)}[/tex]

Cancel the common terms:

[tex]x=\frac{5±\sqrt{5} }{4}[/tex]

Answer:

There is not enough evidence to support the claim that single people who own pets are generally happier than single people without pets. (P-value=0.0509).

Step-by-step explanation:

We have to calculate the difference for every pair, and applied the hypothesis test to this sample of differences.

   Non-pet Pet --> Difference (d)

A 11 13 --> 2

B 9 8 --> -1

C 11 14 --> 3

D 13 13 --> 0

E 6 12 --> 6

F 9 11 --> 2

The claim is that single people who own pets are generally happier than single people without pets. In the context of the sample, it means that the difference between the metric of happiness between the pet and the non-pet subjects is bigger than 0.

The mean and standard deviation of the sample of differences is:

[tex]M=\dfrac{1}{6}\sum_{i=1}^{6}(2+(-1)+3+0+6+2)\\\\\\ M=\dfrac{12}{6}=2[/tex]

[tex]s=\sqrt{\dfrac{1}{(n-1)}\sum_{i=1}^{6}(x_i-M)^2}\\\\\\s=\sqrt{\dfrac{1}{5}\cdot [(2-(2))^2+(-1-(2))^2+(3-(2))^2+(0-(2))^2+(6-(2))^2+(2-(2))^2]}\\\\\\ s=\sqrt{\dfrac{1}{5}\cdot [(0)+(9)+(1)+(4)+(16)+(0)]}\\\\\\ s=\sqrt{\dfrac{30}{5}}=\sqrt{6}\\\\\\s=2.449[/tex]

Then we perform an hypothesis test for the population mean.

The claim is that single people who own pets are generally happier than single people without pets.

Then, the null and alternative hypothesis are:

[tex]H_0: \mu_d=0\\\\H_a:\mu_d> 0[/tex]

The significance level is 0.05.

The sample has a size n=6.

The sample mean is M=2.

As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=2.449.

The estimated standard error of the mean is computed using the formula:

[tex]s_M=\dfrac{s}{\sqrt{n}}=\dfrac{2.449}{\sqrt{6}}=0.9998[/tex]

Then, we can calculate the t-statistic as:

[tex]t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{2-0}{0.9998}=\dfrac{2}{0.9998}=2.0004[/tex]

The degrees of freedom for this sample size are:

[tex]df=n-1=6-1=5[/tex]

This test is a right-tailed test, with 5 degrees of freedom and t=2.0004, so the P-value for this test is calculated as (using a t-table):

[tex]P-value=P(t>2.0004)=0.0509[/tex]

As the P-value (0.0509) is bigger than the significance level (0.05), the effect is  not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that single people who own pets are generally happier than single people without pets.

Answer:

Step-by-step explanation:

Null hypothesis : [tex]H_0:\mu_d=0[/tex]

Alternative hypothesis: [tex]H_0:\mu_d>0[/tex]

Level pf significance [tex]\alpha =0.05[/tex]

Test statistic [tex]t = \frac{\bar d- \mu}{s_d/\sqrt{n} }[/tex]

Mean of difference

[tex]\bar d = \frac{\sum d_i}{n} = 12/6 = 2[/tex]

Standard deviation of difference

[tex]s_d=\sqrt{\frac{\sum (d_i- \bar d)}{n-1} } \\\\=\sqrt{\frac{30}{6.1} } \\\\=2.441[/tex]

Test statistic

[tex]t = \frac{\bar d- \mu}{s_d/\sqrt{n} }[/tex]

[tex]=\frac{2-0}{2.449/\sqrt{6} } \\\\=2.00[/tex]

Degree of freedom

df = n - 1

6-1 = 5

This test is a right-tailed test, with 5 degrees of freedom and t=2.0004, so the P-value for this test is calculated as (using a t-table):

[tex]P-value=P(t>2.0004)=0.0509[/tex]

As the P-value (0.0509) is bigger than the significance level (0.05), the effect is  not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that single people who own pets are generally happier than single people without pets.

Answer:

1: 2.5, 2: 5.0 , 3:7.5 , 4:10 , 5: 12.5

Step-by-step explanation:

asumming it is proportional, you divide 7.5 by 3 which will give you the cost per pound which is $2.5, then you multiply this number by the amount of pounds you want

Answer:

1: 2.5 , 2: 5.0 , 3:7.5 , 4:10 , 5: 12.5

Step-by-step explanation:

Answer:

  x + 0.07x ≥ 30

Step-by-step explanation:

A sum is terms separated by a plus sign.

"Is at least" means "is greater than or equal to."

7% as a decimal fraction is 0.07.

"Of" means "times."

__

In consideration of the above, you have ...

  x  plus  7% of x  is at least  30

  x  +  0.07x  ≥  30 . . . . . using math symbols

Answer: use google calculator It’s reliable

Step-by-step explanation:

Answer: a=1, b=2, c=below x

Step-by-step explanation:

Answer:

Due to the lower z-score of the finishing time, Zandra was faster, that is, doing better in relation to her gender.

Step-by-step explanation:

Z-score:

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question:

Whoever had the lower z-score was faster, that is, did better in relation to their gender.

Roberto:

63.2 minutes. Mean finishing time was 69.4 minutes with a standard deviation of 8.9 minutes. So [tex]X = 63.2, \mu = 69.4, \sigma = 8.9[/tex]

Then

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{63.2 - 69.4}{8.9}[/tex]

[tex]Z = -0.7[/tex]

Zandra:

79.3 minutes. Mean finishing time was 84.7 minutes with a standard deviation of 7.4 minutes. So [tex]X = 79.3, \mu = 84.7, \sigma = 7.4[/tex]

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{79.3 - 84.7}{7.4}[/tex]

[tex]Z = -0.73[/tex]

Due to the lower z-score of the finishing time, Zandra was faster, that is, doing better in relation to her gender.

Answer:

[tex]r_4 = 65[/tex]

Step-by-step explanation:

Given

[tex]r_1 = 2[/tex]

[tex]r_n = 4r_{n-1} - 3[/tex]

Required

Find [tex]r_4[/tex]

Calculating the value of [tex]r_4[/tex]

This means n = 4;

Hence,

[tex]r_n = 4r_{n-1} - 3[/tex]

[tex]r_4 = 4r_{4-1} - 3[/tex]

[tex]r_4 = 4r_3 - 3[/tex]

At this point, we need to solve for [tex]r_3[/tex];

Taking n as 3

[tex]r_n = 4r_{n-1} - 3[/tex]

[tex]r_3 = 4r_{3-1} - 3[/tex]

[tex]r_3 = 4r_2 - 3[/tex]

At this point, we need to solve for [tex]r_2[/tex];

Taking n as 2

[tex]r_n = 4r_{n-1} - 3[/tex]

[tex]r_2 = 4r_{2-1} - 3[/tex]

[tex]r_2 = 4r_1 - 3[/tex]

Substitute 2 for [tex]r_1[/tex]

[tex]r_2 = 4 * 2 - 3[/tex]

[tex]r_2 = 8 - 3[/tex]

[tex]r_2 = 5[/tex]

Solving for [tex]r_3[/tex]

Substitute 5 for [tex]r_2[/tex]

[tex]r_3 = 4 * 5 - 3[/tex]

[tex]r_3 = 20 - 3[/tex]

[tex]r_3 = 17[/tex]

Solving for [tex]r_4[/tex]

Substitute 17 for [tex]r_3[/tex]

[tex]r_4 = 4 * 17 - 3[/tex]

[tex]r_4 = 68 - 3[/tex]

[tex]r_4 = 65[/tex]

Hence, [tex]r_4 = 65[/tex]

Answer:

1) This is because some has their own shape and the commonly used loop is the clothoid loop, e.t.c.

2) This is because researchers found out that it helps ease the passage of the kidney stones.

Step-by-step explanation:

Hope it helps

Thanks.

Answer:

1) Roller coaster loops are never circular...why do you think that is?:

This force is called the Centripetal Force.

Newton's first Law of Motion of motion tells us that, without this force, the coaster would like to travel in a straight line and at constant speed. The centripetal force is pushing the coaster around in a circle. It is your body's (equal and opposite) reaction to this force, often referred to as the Centrifugal Force, that explains the feeling you get .There is a force (provided by the rails), that is pushing the trucks of the coaster towards the centre of the loop. t of being squashed into your seat (Newton's third Law of Motion)

2. Riding Big Thunder Mountain Railroad at Disney World could help dislodge kidney stones....why do you think that is?:

Riding the Big Thunder Mountain Railroad roller coaster at Disney World could help ease the passage of small kidney stones.  Kidney stones are hard masses of minerals that form in the kidneys. They can range in size, from a tiny grain of sand to, in extreme cases, the size of a golf ball. Patients with kidney stones don't always need treatment, because the stones can pass out of the body on their own, but the process of passing them can be quite painful. Riding the Big Thunder Mountain Railroad roller coaster at Disney World could help ease the passage of small kidney stones, according to the new study because of 'The intense ride helps get rid of them'.

The median is the middle number in the data set when

the data set is written from least to greatest.

So let's write our data set from least to greatest.

4, 9, 15, 16, 18, 21, 25, 27, 30, 31, 33

Now, we identify the middle number.

4, 9, 15, 16, 18, 21, 25, 27, 30, 31, 33

Notice that 21 appears in the middle.

So the median of the data set is 21.

Answer:

Median: 21

Step-by-step explanation:

arrange all the numbers from least to greatest:

4, 9, 15, 16, 18, 21, 25, 27, 30, 31, 33

median = is the middle number in the data set/listed numbers.

Since there are eleven numbers in the set of data, we divide it equally, to find the median.

{4, 9, 15, 16, 18} 21 {25, 27, 30, 31, 33}

Step-by-step explanation:

Second option is the correct answer

Answer:

B

Step-by-step explanation:

Area of sector = m/360(πr²)

Answer:

4

Step-by-step explanation:

Answer:

Step-by-step explanation:

A=18000(1+0.02)^11

A=22,380.74

Answer:

y≈22828

Step-by-step explanation:

Answer:

the value of x is x=1

Step-by-step explanation:

3x=3

x=3÷3

x=1

1.

Step-by-step explanation:

Answer:

(a) The sample sizes are 6787.

(b) The sample sizes are 6666.

Step-by-step explanation:

(a)

The information provided is:

Confidence level = 98%

MOE = 0.02

n₁ = n₂ = n

[tex]\hat p_{1} = \hat p_{2} = \hat p = 0.50\ (\text{Assume})[/tex]

Compute the sample sizes as follows:

[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{2\times\hat p(1-\hat p)}{n}[/tex]

       [tex]n=\frac{2\times\hat p(1-\hat p)\times (z_{\alpha/2})^{2}}{MOE^{2}}[/tex]

          [tex]=\frac{2\times0.50(1-0.50)\times (2.33)^{2}}{0.02^{2}}\\\\=6786.125\\\\\approx 6787[/tex]

Thus, the sample sizes are 6787.

(b)

Now it is provided that:

[tex]\hat p_{1}=0.45\\\hat p_{2}=0.58[/tex]

Compute the sample size as follows:

[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{\hat p_{1}(1-\hat p_{1})+\hat p_{2}(1-\hat p_{2})}{n}[/tex]

       [tex]n=\frac{(z_{\alpha/2})^{2}\times [\hat p_{1}(1-\hat p_{1})+\hat p_{2}(1-\hat p_{2})]}{MOE^{2}}[/tex]

          [tex]=\frac{2.33^{2}\times [0.45(1-0.45)+0.58(1-0.58)]}{0.02^{2}}\\\\=6665.331975\\\\\approx 6666[/tex]

Thus, the sample sizes are 6666.

Answer:

140.4 yards squared

Step-by-step explanation:

Answer:

Step-by-step explanation:

Hello!

Full Text:

As part of its "It Can Wait" campaign to discourage texting while driving, AT&T recently released the results of separate surveys concerning the extent of texting while driving among adults twd_commutor_survey and among teens att_teen_survey_executive . In the survey of the adults, 496 of n1 = 1,011 adult drivers (49.06%) said they text while driving. In the survey of the teens, 516 of n2 = 1,200 (43%) said they text while driving.

Question 1. Calculate a 98% confidence interval for the difference padult - pteen in the proportions of adults that text while driving and the proportion of teens that text while driving. (use 4 decimal places in your answers) lower bound upper bound

Be:

X₁: Number of adult drivers that text while driving, out of 1011.

n₁= 1011

x₁= 496

p₁'= 496/1011= 0.4906

X₂: Number of teen drivers that text while driving, out of 1200.

n₂= 1200

x₂= 516

p₂'= 516/1200= 0.43

For the 98% CI for p₁-p₂

[tex]Z_{1-\alpha /2}= Z_{0.99}= 2.326[/tex]

(p₁'-p₂')±[tex]Z_{1-\alpha /2}[/tex] * [tex]\sqrt{\frac{p'_1(1-p'_1)}{n_1} +\frac{p'_2(1-p'_2)}{n_2} }[/tex]

(0.4906-0.43)±2.326*[tex]\sqrt{\frac{0.4906(1-0.4906)}{1011} +\frac{0.43(1-0.43)}{1200} }[/tex]

0.0606±2.326*0.0212

[0.011; 0.11]

Using a 98% confidence level, you'd expect that the interval  [0.011; 0.11] contains the difference between the population proportion of adults that text while driving and the population proportion of teens that ext while driving.

Question 2. Choose the correct interpretation of the above confidence interval. Note: 2 submissions allowed.

To decide over a hypothesis test using a confidence interval there are several conditions that should be met:

1) The hypotheses should be two-tailed:

H₀: p₁ - p₂= 0

H₁: p₁ - p₂≠ 0

2) The confidence level of the interval and the significance level of the test should be complementary, this means that if the interval was constructed with a level 1 - α: 0.98 then the test should be made using α: 0.02.

Naturally, the hypotheses and the CI should be made for the same parameters.

If all conditions are met, the decision criteria is as follows:

If the CI contains the value stated in the null hypothesis, the decision is to not reject the null hypothesis.

If the CI doesn't contain the value stated in the null hypothesis, the decision is to reject the null hypothesis.

In this case, the value stated in the null hypothesis is "zero" and is not included in the interval, so the decision is to reject the null hypothesis. You can conclude that the population proportions of adults and teens that text while driving are different.

Considering that the CI is positive, we can think that the proportion of adults that text while driving is grater than the proportion of teens that text while driving.

Options:

1) Since 0 is not in the confidence interval, the surveys provide evidence that the proportion of teens that text while driving is greater than the proportion of adults that text while driving.

2) Since the confidence interval is entirely positive, the surveys provide evidence that the proportion of teens that text while driving is greater than the proportion of adults that text while driving.

3) Since the confidence interval is entirely positive, the surveys provide evidence that the proportion of adults that text while driving is greater than the proportion of teens that text while driving.

4) Since 0 is not in the confidence interval, the surveys provide evidence that there is no significant difference between the proportions of adults and teens that text while driving.

The correct option is: "3"

I hope this helps!

Answer:

• The base is 8 and the height is 6

Step-by-step explanation:

A rectangle has two dimensions:

The base, which is the distance between the points who have the same value of y.

The height, which is the distance between the points who have the same value of x.

Distance between 2 points:

Points (a,b) and (c,d).

[tex]D = \sqrt{(c-a)^{2} + (d-b)^{2}}[/tex]

I suppose there was a small typing mistake, as these points do not make a rectangle.

I will say that we have these following points:

L(0,6), M(8,6), N(8,0), O(0,0).

Base:

Same value of y.

L(0,6), M(8,6), or N(8,0) and O(0,0).

They will have the same result, will use the second.

[tex]D = \sqrt{(8-0)^{2} + (0-0)^{2}} = \sqrt{64} = 8[/tex]

The base is 8.

Height:

Same value of x.

L(0,6), O(0,0) or M(8,6) and N(8,0).

[tex]D = \sqrt{(8-8)^{2} + (6-0)^{2}} = \sqrt{36} = 6[/tex]

The height is 6.

So the correct answer is:

• The base is 8 and the height is 6

Answer:

69

Step-by-step explanation:

first 135

second 135 - 20% = 135 - 27 = 108

third 108 - 20% = 108 - 21.6 = 86.4

fourth 86.4 -20% = 86.4 -17.28= 69.12

rounded 69

Why is this?

We start with x^4 - x^2, which is the original f(x) function. Adding some number to this result will increase the y coordinate of any point on the f(x) function. This is because y = f(x). The only thing that matches is choice C, where we shift the graph up 0.5 units. We say that g(x) = f(x) + 0.5

Choice D goes in the opposite direction, and shifts the graph down 0.5 units.

Choices A and B shift the graph horizontally to the right 0.5 units and to the left 0.5 units respectively.

Answer:

She had at least 21

Step-by-step explanation:

She had at least 4+ 12 +5  

She had at least 21

She spent 4 and 12 so she had 16 dollars

She had at least 5 left

so at the very least she had 16+5 which is 21, she may have had more

Answer:

21

Step-by-step explanation:

she spent 4 and 12 so she had 16 dollars so she have at least she have $5 left

Here is the complete question.

A one-tailed hypothesis test with the t statistic Antisocial personality disorder (ASPD) is characterized by deceitfulness, reckless disregard for the well-being of others, a diminished capacity for remorse, superficial charm, thrill seeking, and poor behavioral control. ASPD is not normally diagnosed in children or adolescents, but antisocial tendencies can sometimes be recognized in childhood or early adolescence. James Blair and his colleagues have studied the ability of children with antisocial tendencies to recognize facial expressions that depict sadness, happiness, anger, disgust, fear, and surprise. They have found that children with antisocial tendencies have selective impairments, with significantly more difficulty recognizing fearful and sad expressions. Suppose you have a sample of 40 16-year-old children with antisocial tendencies and you are particularly interested in the emotion of fear. The average 16-year-old has a score on the emotion recognition scale of 11.80. (The higher the score on this scale, the more strongly an emotion has to be displayed to be correctly identified. Therefore, higher scores indicate greater difficulty recognizing the emotion). Assume that scores on the emotion recognition scale are normally distributed. You believe that children with antisocial tendencies will have a harder time recognizing the emotion of fear in other words, they will have higher scores on the emotion recognition test).

What is your null hypothesis stated using symbols?

What is your alternative hypothesis stated using symbols?

This is a               tailed test. Given what you know, you will evaluate this hypothesis using a               statistic.

Answer:

Step-by-step explanation:

Given that :

the population mean = 11.80

Thus;

The null hypothesis stated using symbols is :

[tex]\mathbf{H_o = \mu = 11.80}[/tex]

The alternative hypothesis stated using symbols is:

[tex]\mathbf{H_i = \mu > 11.80}[/tex]

However, Since the alternative hypothesis looks somewhat greater than the null hypothesis, then the test is based one right - tailed test

Thus;

This is a     one   tailed test and the hypothesis uses a    t   statistic

Answer:

Step-by-step explana[tex]f(x) = (x^{2} - 6x + 36) . (x^{2} + 2x + 4)\\f (x) = (x^{4} + 2x^{3} + 4x^{2} - 6x^{3} - 12x^{2} - 24x + 36x^{2} + 72x + 144)\\f(x) = (x^{4} - 4x^{3} + 28x^{2} + 48x + 144)[/tex]

x=6 and x=-2 are the roots of the equation f(x)= (x - 6)²(x + 2)².

A quadratic equation is a second-order polynomial equation in a single variable x , ax²+bx+c=0. with a ≠ 0 .

We need to find the roots of f(x) = (x - 6)²(x + 2)²

f(x)= (x - 6)²(x + 2)²

f of x equal to x minus six whole power six into x plus two whole square

=(x²+36-12x)(x²+4+4x)

Apply distributive law.

=x⁴+4x²+4x³+36x²+144+144x-12x³-48x-48x²

=x⁴-8x³-8x²+96x+144

Hence, x=6 and x=-2 are the roots of the equation f(x)= (x - 6)²(x + 2)².

To learn more on Quadratic equation click:

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Answer:

Step-by-step explanation:

Short answer: area is maximized when half the cost is spent in each of the orthogonal directions. This means the east and west sides will total $350 at $20 per foot, so will be 17.5 feet. The north and south sides will total $350 at $9 per foot, so will be 38 8/9 feet.

The dimensions that maximize the area are 17.5 ft in the north-south direction by 38 8/9 ft in the east-west direction.

__

Long answer: If x represents the length of the north and south sides, and y represents the length of the east and west sides, then the total cost is ...

  10y +10y +2x +7x = 700

  9x +20y = 700

  y = (700 -9x)/20

We want to maximize the area:

  A = xy = x(700 -9x)/20

We can do this by differentiating and setting the derivative to zero:

  dA/dx = 700/20 -9x/10 = 0

  350 -9x = 0 . . . . multiply by 10

  x = 350/9 = 38 8/9

  y = (700 -9(350/9))/20 = 350/20 = 17.5

The north and south sides are 38 8/9 ft long; the east and west sides are 17.5 ft long to maximize the area for the given cost.

Answer:

The fourth term of the expansion is -220 * x^9 * y^3

Step-by-step explanation:

Question:

Find the fourth term in (x-y)^12

Solution:

Notation: "n choose k", or combination of k objects from n objects,

C(n,k) = n! / ( k! (n-k)! )

For example, C(12,4) = 12! / (4! 8!) = 495

Using the binomial expansion formula

(a+b)^n

= C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + C(n,3)a^(n-3)b^3 + C(n,4)a^(n-4)b^4 +....+C(n,n)b^n

For (x-y)^12, n=12, k=3, a=x, b=-y, and the fourth term is

C(n,3)a^(n-3)b^3

=C(12,3) * x^(12-3) * (-y)^(3)

= 220*x^9*(-y)^3

= -220 * x^9 * y^3

The fourth term in the binomial expansion of ( x - y )¹² is given by the equation A₄ = -220 x⁹y³

The general term of the binomial expansion is Tr+1 = nCr x^n-r y^r . Here the coefficient values are found from the pascals triangle or using the combinations formula, and the sum of the exponents of both the terms in the general term is equal to n.

( x + y )ⁿ = ⁿCₐ ( x )ⁿ⁻ᵃ ( y )ᵃ

Given data ,

Let the binomial expansion be represented as A

Now , the value of A is

A = ( x - y )¹²   be equation (1)

On simplifying the equation , we get

The fourth term of the binomial expansion is calculated by

A₄ = ⁿC₃ a⁽ⁿ⁻³⁾ b³

Substituting the values in the equation , we get

A₄ = ¹²C₃ x⁽¹²⁻³⁾ ( -y )³

On further simplification , we get

A₄ = ( 12 )! / ( 9 )! 3! x⁹ ( -y )³

A₄ = 12 x 11 x 10 / 2 x 3 x⁹ ( -y) ³

A₄ =  -220 x⁹y³

Hence , the fourth term of binomial expansion is  -220 x⁹y³

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