ANSWER
4,410
EXPLANATION
The number of ways we can choose 4 teachers from 7 teachers is,
[tex]_7C_4=\frac{7!}{(7-4)!\times4!}=\frac{7\times6\times5\times4!}{3!\times4!}=\frac{7\times6\times5}{3\times2}=\frac{7\times6\times5}{6}=7\times5=35[/tex]There are 35 ways of choosing 4 teachers out of 7.
And the number of ways we can choose 4 students from 9 students is,
[tex]\begin{gathered} _9C_4=\frac{9!}{(9-4)!\times4!}=\frac{9\times8\times7\times6\times5\times4!}{5!\times4!}=\frac{9\times8\times7\times6\times5}{5\times4\times3\times2} \\ _9C_4=\frac{9\times8\times7}{4}=\frac{9\times(2\times4)\times7}{4}=9\times7\times2=126 \end{gathered}[/tex]There are 126 ways of choosing 4 students out of 9.
The committee is formed by 4 teachers and 4 students. The number of ways it can be made is,
[tex]_7C_4\times_9C_4=35\times126=4,410[/tex]Hence, there are 4,410 ways to choose 4 students and 4 teachers out of 9 students and 7 teachers.
can you please give me any examples on how to do this
we can take two numbers of the sequence and subtract them to see the difference
so
[tex]1.9-1.2=0.7[/tex]the sequence adds 0.7 each step
the next 3 terms are
[tex]3.3+0.7=4[/tex][tex]4+0.7=4.7[/tex][tex]4.7+0.7=5.4[/tex]xyx2xy1645256720 2258 484 1,2762873 7842,044 3294 1,0243,008 45141 2,025 6,345 ∑x=143 ∑y=411 ∑x2=4,573 ∑xy=13,393 Which regression equation correctly models the data?y = 2.87x + 0.12y = 2.87x + 11.85y = 3.39x – 14.75y = 3.39x – 9.24
We are asked to identify the correct regression equation.
The regression equation is given by
[tex]y=bx+a[/tex]Where the coefficients a and b are
[tex]a=\frac{\sum y\cdot\sum x^2-\sum x\cdot\sum xy}{n\cdot\sum x^2-(\sum x)^2}[/tex][tex]b=\frac{n\cdot\sum xy-\sum x\cdot\sum y}{n\cdot\sum x^2-(\sum x)^2}[/tex]Where n is the number of observations that is 5.
Let us substitute the following into the above formula.
∑x=143
∑y=411
∑x^2=4,573
∑xy=13,393
[tex]a=\frac{411\cdot4573-143\cdot13393}{5\cdot4573-(143)^2}=-14.75[/tex][tex]b=\frac{5\cdot13393-143\cdot411}{5\cdot4573-(143)^2}=3.39[/tex]So, the coefficients are
a = -14.75
b = 3.39
Therefore, the correct regression equation is
[tex]y=3.39x-14.75[/tex]Sara has saved $500 and wants to buy a new computer. the computer she wants costs $1400. her current job pays $17.50 per hour after taxes . use an inequality to describe the situation , solve the inequality and write a sentence describing what the solution means to Sara
She has already 500
x hours woked
pund we got:
7.50 x >= 900
And inequality could be
fx:o
[tex]500+17.5x\ge1400[/tex]because we need to gain 1400 or more
hours in order to have enough money to buy the computer
Sara needs to work at least 52 00/17.5
x >= 5r t o
The equatu
1
And then we can solve for x a
500+17.5x>>= 1400
utioen x
For this50 case we can do this:
500+ 17.
Describe the association in the scatter plot below.----------------The scatter plot shows (positive linear, positive linear with one outlier, negative linear, negative linear with one outlier, nonlinear, or no) association because as the plotted values of x increase, the values of y generally (decrease, increase, show no pattern or follow a nonlinear pattern).
From the given figure
The given point can form a line with a negative slope, because when
the values of x increase the values of y decrease
Then the scatter plot shows a negative linear association because
as the values of x increase, the values of y generally decrease
O DESCRIPTIVE STATISTICInterpreting relative frequency-histogramsStudents at a major university in Southern California are complaining about a serious housing crunch. Many of the university's students, they complain, have tocommute too far to school because there is not enough housing near campus. The university officials' response is to perform a study. The study, reported in theschool newspaper, contains the following histogram summarizing the commute distances for a sample of 100 students at the university:Relative frequencyCommute distance (in miles)Based on the histogram, find the proportion of commute distances in the sample that are at least 16 miles. Write your answer as a decimal, and do not roundyour answer
Since the graph gives us the relative frequency we just have to add those who are more or equal to 16; in this case we have to add 0.11 and 0.06, therefore the proportion in this case is 0.17
Determine whether the ratios are equivalent.
2:3 and 24:36
O Not equivalent O Not equivalent
We can conclude that the given ratios 2:3 and 24:36 are equivalent.
What are ratios?A ratio in math displays how many times one number is contained in another. The ratio of oranges to lemons, for instance, is eight to six if there are eight oranges and six lemons in a bowl of fruit. The proportions of oranges to the total amount of fruit are 8:14 for oranges and 6:8 for lemons, respectively.So, ratios are equivalent or not:
2:3 and 24:362/3 = 24/362/3 = 2/3 (Divide by 12)Then, 2:3 :: 2:3
Therefore, we can conclude that the given ratios 2:3 and 24:36 are equivalent.
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Which recipe makes more cookies per cup of chocolate chips?
We are given the recipes of Grandma and Betty Potter box.
We are asked to find out which of them makes more cookies per cup of chocolate chips.
Let us find the unit rate for both of them and compare which is greater.
Grandma:
She uses 1 1/2 cups of chocolate chips to make 24 cookies.
The unit rate is
[tex]\frac{24}{1\frac{1}{2}}=\frac{24}{\frac{3}{2}}=24\times\frac{2}{3}=16[/tex]So the unit rate is 16 cookies per chocolate chip.
Betty Potter box:
60 cookies are made using 3 chocolate chips.
80 cookies are made using 4 chocolate chips.
200 cookies are made using 10 chocolate chips.
The unit rate is
[tex]\begin{gathered} \frac{60}{3}=20 \\ \frac{80}{4}=20 \\ \frac{200}{10}=20 \end{gathered}[/tex]So the unit rate is 20 cookies per chocolate chip.
As you can see, the recipe of Better Potter box makes more cookies per cup of chocolate chip.
3(4x+1)^2-5=25 using square root property
Answer:
[tex]x=\frac{-1+\sqrt{10}}{4}\text{ or }x=\frac{-1-\sqrt{10}}{4}[/tex]Explanation:
Given the equation:
[tex]3\left(4x+1\right)^2-5=25[/tex]To solve an equation using the square root property, begin by isolating the term that contains the square.
[tex]\begin{gathered} 3(4x+1)^{2}-5=25 \\ \text{ Add 5 to both sides of the equation} \\ 3(4x+1)^2-5+5=25+5 \\ 3(4x+1)^2=30 \\ \text{ Divide both sides by 3} \\ \frac{3(4x+1)^2}{3}=\frac{30}{3} \\ (4x+1)^2=10 \end{gathered}[/tex]After isolating the variable that contains the square, take the square root of both sides and solve for the variable.
[tex]\begin{gathered} \sqrt{(4x+1)^2}=\pm\sqrt{10} \\ 4x+1=\pm\sqrt{10} \\ \text{ Subtract 1 from both sides} \\ 4x=-1\pm\sqrt{10} \\ \text{ Divide both sides by 4} \\ \frac{4x}{4}=\frac{-1\pm\sqrt{10}}{4} \\ x=\frac{-1\pm\sqrt{10}}{4} \end{gathered}[/tex]Therefore, the solutions to the equation are:
[tex]x=\frac{-1+\sqrt{10}}{4}\text{ or }x=\frac{-1-\sqrt{10}}{4}[/tex]
A statement of Chandler's biweekly earnings is given below. What is Chandler's gross pay?
SOLUTION:
Step 1:
In this question, we are asked to calculate Chandler's gross pay from the statement of bi-weekly earnings.
Step 2:
To get the Gross pay, we need to do the following:
[tex]\text{Gross pay - Total Deductions = Net Pay}[/tex]Now, we need to calculate Total Deductions:
[tex]\text{ \$ 105.00 + \$ 52.14 + \$ 10.62 + \$ 26. 15 = \$ 193.91}[/tex]Now, we have that the Net Pay = $ 780. 63
Then,
[tex]\begin{gathered} \text{Gross Pay - \$ 193. 91 = \$ 7}80.\text{ 63} \\ \text{Gross pay = \$ 780.63 + \$ 193.91} \\ \text{Gross Pay = \$ 974. 54} \end{gathered}[/tex]CONCLUSION:
Chandler's Gross Pay = $ 974. 54
Describe a series of transformations that takes triangle ABC to triangle A’B’C’
Notice that if the triangle ABC is reflected over the X axis (red), and then reflected over the Y axis (green), we just would have to translate the triangle two units to the right (blue) to get A'B'C':
Which of the following numbers are greater than 6 and less than 8? Explainhow you know.
Let's analyze each case and see if they are less than 8 first:
[tex]7<8,therefore(7)^{\frac{1}{2}}<8[/tex][tex]\sqrt[]{64}=8[/tex]thus
[tex]\sqrt[]{60}<\sqrt[]{64}=8[/tex][tex]\sqrt[]{60}<8[/tex]Finally,
[tex]64<80[/tex][tex]\sqrt[]{64}<\sqrt[]{80}[/tex]thus
[tex]8<\sqrt[]{80}[/tex]In sumary, the first two options are less than 8, but not the third.
Now let's see if they are greater than 6
[tex]9>7[/tex][tex]\sqrt[]{9}>\sqrt[]{7}[/tex][tex]3>\sqrt[]{7}[/tex]and
[tex]6>3>\sqrt[]{7}[/tex]thus
[tex]6>\sqrt[]{7}[/tex]Now
[tex]36<60[/tex][tex]\sqrt[]{36}<\sqrt[]{60}[/tex]and so
[tex]6<\sqrt[]{60}[/tex]Finally
[tex]\sqrt[]{80}>\sqrt[]{60}>6[/tex]thus
[tex]\sqrt[]{80}>6[/tex]In conclussion, the second and third options are greater than 6, but not the first.
A data set is summarized in the frequency table below. Using the table, determine the number of values less than or equal to 6.ValueFrequency152332435463788397108113Give your answer as a single number. For example if you found the number of values was 14, you would enter 14.
The number of values less than or equal to 6 is 5 + 3 +2 +3 +4 +3 = 20
the discriminant equation How many real solution 4x^2-8x+10=-x^2-5 have?
Answer:
0 real solutions
Explanation:
First, we need to transform the equation into the form:
[tex]ax^2+bx+c=0[/tex]So, the initial equation is equivalent to:
[tex]\begin{gathered} 4x^2-8x+10=-x^2-5 \\ 4x^2-8x+10+x^2+5=-x^2-5+x^2+5 \\ 5x^2-8x+15=0 \end{gathered}[/tex]Now, the discriminant can be calculated as:
[tex]b^2-4ac[/tex]If the discriminant is greater than 0, the equation has 2 real solutions.
If the discriminant is equal to 0, the equation has 1 real solution
If the discriminant is less than 0, the equation has 0 real solutions
So, in this case, a is 5, b is -8 and c is 15. Then, the discriminant is equal to:
[tex](-8)^2-4\cdot5\cdot15=84-300=-236[/tex]Since the discriminant is less than zero, the equation has 0 real solutions
which methods correctly solve for the variable x in the equation 2/5m = 8?
Ok, so the equation is (2/5)m=8
1st option: Divide by 2 on both sides, then multiply by 5 on both sides:
[tex]\begin{gathered} \frac{2}{10}m=4 \\ \frac{10}{10}m=20 \\ m=20 \end{gathered}[/tex]2nd option: Multiply both sides by 5/2
[tex]\begin{gathered} \frac{2}{5}\cdot\frac{5}{2}m=8\cdot\frac{5}{2} \\ m=20 \end{gathered}[/tex]3rd option: First dristibute 2/5 to (m=8), the multiply by 5/2 in both sides
[tex]\begin{gathered} \frac{2}{5}m=8 \\ \frac{5}{2}\cdot\frac{2}{5}m=8\cdot\frac{5}{2} \\ m=20 \end{gathered}[/tex]4th option: Divide both sides by 2/5:
[tex]\begin{gathered} \frac{\frac{2}{5}}{\frac{2}{5}}m=8\cdot\frac{5}{2} \\ m=20 \end{gathered}[/tex]5th option: First, multiply by 5. Then, divide by 2.
[tex]\begin{gathered} 5\cdot\frac{2}{5}m=40 \\ 2m/2=40/2 \\ m=20 \\ \end{gathered}[/tex]All the methods are correct
In the figure to the right, ABC and ADE are similar. Find the length of EC.
The length of EC is ___.
Answer:
ninety 90 feet or foot long
consider the parent function f(x)=x^2. a. graph y=f(x). b. write an equation for f(1/2x). Then sketch a graph of y=f(1/2x) and describe the transformation. c.write an equation for f(3x). Then sketch a graph of y=f(3x) and describe the transformation.
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
f(x)=x²
f(1/2x) = ?
graph of y=f(1/2x) = ?
Step 02:
b. f(1/2x)
x ===> 1/2x
[tex]f\text{ (1/2 x) = (}\frac{1}{2}x)^2=\frac{1}{4}x^2[/tex]Step 03:
c. Graph:
We give values to x, and we obtain the values of y.
f(x) = 1/4 x²
e.g.
if x = 4
y = 1/4 (4)² = 1/4 * 16 = 4
That is the solution for b. and c.
The sum of 5 times a number and 7 equals 8. Find the number
Explanation
Let the number be x. Therefore, we will have
[tex]\begin{gathered} 5x+7=8 \\ 5x=8-7 \\ 5x=1 \\ x=\frac{1}{5} \end{gathered}[/tex]NO LINKS!! Please help me with this problem
0.3821, 0.8745
========================================================
Work Shown:
pi/2 = 3.14/2 = 1.57 approximately
The solutions for t must be in the interval 0 ≤ t ≤ 1.57
[tex]3\cos(5t)+3 = 2\\\\3\cos(5t) = 2-3\\\\3\cos(5t) = -1\\\\\cos(5t) = -1/3\\\\5t = \cos^{-1}(-1/3)\\\\5t \approx 1.9106+2\pi n \ \text{ or } \ 5t \approx -1.9106+2\pi n\\\\t \approx \frac{1.9106+2\pi n}{5} \ \text{ or } \ t \approx \frac{-1.9106+2\pi n}{5}\\\\[/tex]
where n is an integer.
Let
[tex]P = \frac{1.9106+2\pi n}{5}\\\\Q = \frac{-1.9106+2\pi n}{5}\\\\[/tex]
Then let's generate a small table of values like so
[tex]\begin{array}{|c|c|c|} \cline{1-3}n & P & Q\\\cline{1-3}-1 & -0.8745 & -1.6388\\\cline{1-3}0 & **0.3821** & -0.3821\\\cline{1-3}1 & 1.6388 & **0.8745**\\\cline{1-3}2 & 2.8954 & 2.1312\\\cline{1-3}\end{array}[/tex]
The terms with surrounding double stars represent items in the interval 0 ≤ t ≤ 1.57
Therefore, we end up with the solutions 0.3821 and 0.8745 both of which are approximate.
You can use a graphing tool like Desmos or GeoGebra to verify the solutions. Be sure to restrict the domain to 0 ≤ t ≤ 1.57
Answer:
[tex]\textsf{c)} \quad 0.3821, \; 0.8745[/tex]
Step-by-step explanation:
Given equation:
[tex]3 \cos (5t)+3=2, \quad \quad 0\leq t\leq \dfrac{\pi}{2}[/tex]
Rearrange the equation to isolate cos(5t):
[tex]\begin{aligned}\implies 3 \cos(5t)+3&=2\\3 \cos(5t)&=-1\\\cos(5t)&=-\dfrac{1}{3}\end{aligned}[/tex]
Take the inverse cosine of both sides:
[tex]\implies 5t=\cos^{-1}\left(-\dfrac{1}{3}\right)[/tex]
[tex]\implies 5t=1.91063..., -1.91063...[/tex]
As the cosine graph repeats every 2π radians, add 2πn to the answers:
[tex]\implies 5t=1.91063...+2\pi n, -1.91063...+2 \pi n[/tex]
Divide both sides by 5:
[tex]\implies t=0.38212...+\dfrac{2}{5}\pi n,\;\; -0.38212...+\dfrac{2}{5} \pi n[/tex]
The given interval is:
[tex]0\leq t\leq \dfrac{\pi}{2}\implies0\leq t\leq 1.57079...[/tex]
Therefore, the solutions to the equation in the given interval are:
[tex]\implies t=0.3821, \; 0.8745[/tex]
A bag contains 8 red marbles, 7 blue marbles and 6 green marbles. If three marbles are drawn out of the bag without replacement, what is the probability, to the nearest 10th of a percent, that all three marbles drawn will be red?
SOLUTION
Given the question, the following are the solution steps to answer the question.
STEP 1: Write the formula for probability
[tex]Probability=\frac{number\text{ of required outcomes}}{number\text{ of total possible outcomes}}[/tex]STEP 2: Write the outcomes of the events
[tex]\begin{gathered} number\text{ of red marbles}\Rightarrow n(red)\Rightarrow8 \\ number\text{ of blue marbles}\Rightarrow n(blue)\Rightarrow7 \\ number\text{ of green marbles}\Rightarrow n(green)\Rightarrow6 \\ number\text{ of total marbles}\Rightarrow n(total)\Rightarrow21 \end{gathered}[/tex]STEP 3: Write the formula for getting the probability that all three marbles drawn will be red
[tex]Pr(Red\text{ and Red and Red\rparen}\Rightarrow Pr(red)\times Pr(red)\times Pr(red)[/tex]STEP 4: Calculate the probability
[tex]\begin{gathered} Pr(all\text{ three are reds\rparen}\Rightarrow\frac{8}{21}\times\frac{7}{20}\times\frac{6}{19} \\ =\frac{336}{7980}=0.042105263 \\ To\text{ percentage will be to multiply by 100} \\ 4.210526316\% \\ To\text{ the nearest tenth will be:} \\ \approx4.2\% \end{gathered}[/tex]Hence, the probability, to the nearest 10th of a percent, that all three marbles drawn will be red is 4.2%
The diameter of circle is 20 inches. find the circumference in terms of pi
The below formula is used to find the circumference of a circle;
[tex]C=2\pi r[/tex]But we know that the diameter of a circle is expressed as;
[tex]d=2r[/tex]Let's replace 2r with d in the 1st equation, we'll then have;
[tex]C=\pi d[/tex]We've been told that the diameter of the circle is 20inches, if we substitute this value into our equation, we'll have;
[tex]C=20\pi[/tex]Which expression is undefined? O A. 11 B.- 3 C.6-6) D. -4+0
Answer:
Option C
Step-by-step explanation:
Undefined expression:
Division by 0, or fraction in which the denominator is 0. In this question, this is in option C, since 3/(6-6) = 3/0.
Solve the following system of equations by graphing3x+5y=10y=-x+4
ANSWER
The point of intersection of the two equations is (5, - 1)
The graph is
STEP BY STEP EXPLANATION
Step 1: The given equations are:
3x + 5y = 10
y= -x + 4
Step 2: Assume values for x in a table (example -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5) to determine the corresponding values for y for both equations
Step 3: Graph the equations and locate the intersection of the two equations
23 – 4u < 11 what is the answer
23 - 4u < 11
23 is adding on the left, then it will subtract on the right
-4u < 11 - 23
-4u < -12
-4 is multiplying on the left, then it will divide on the right. Remember that dividing by a negative number changes the sign.
u > (-12)/(-4)
u > 3
In basketball, " one on one" free throw shooting ( commonly called foul shooting) is done as follows: if the player makes the first shot(1point), she is given a second shot. If she misses the first shot, she is not given a second shot. Christine, a basketball player, has a 70% free throw record. (she makes 70% of her free throws). Find the probability that, given one-on-one free throw shooting opportunity, Christene will score one point.
If she will be able to shoot the first shot and miss the second shot, then she will obtain 1 point.
Thus, the probability that Christine will get the first shot is as follows:
[tex]P(1pt)=(0.7)(0.3)=0.21[/tex]where the first factor is the probability that she will shoot the first shot and the second factor is the probability that she missed the second shot. Thus, the probability of obtaining 1 point is 21% or 0.21.
Zales sells diamonds for $1,100 that cost $800. What is Zales’s percent markup on selling price? Check the selling price.
Zales's percent markup on the selling price as required in the task content is; 37.5%.
Percentages and markup priceIt follows from the task content that the percent markup on the selling price be determined according to the given data.
Since the cost of diamonds is; $800 while the diamonds sell for $1,100. It follows that the markup on the selling price of the diamonds is;
Markup = Selling price - Cost price.
Hence, we have;
Markup = 1,100 - 800.
Therefore, the markup is; $300.
On this note, the percent markup can be determined as follows;
= (300/800) × 100%.
= 37.5%.
Ultimately, the percent markup on the diamonds is: 37.5%.
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in the experiment of the preceding exercise, the subjects were randomly assigned to the different treatments. what is the most important reason for this random assignment?
The most important reason for random assignment on the subjects in the experiment, is because random assignment would be the best way in creating group of subjects to the different treatments.
Note that; the group of subjects are roughly equivalent at the beginning of the experiment.
Using random assignment will allow the allocation of different patients to various treatments at a random order. From this there will be objective results obtained altogether from the experiment under investigation.
Random assignment will eliminate any biasness that may occur when conducting the experiment. It prevents favoritism of any event from occurring. It will ensure that all the different patients have an equal chance of being selected for various treatment.
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let f ( x ) = 6356 x + 5095 . Use interval notation. Many answers are possible.
The equation of the function has its domain representation in interval notation as (oo, oo)
How to determine the domain of the functionFrom the question, the equation of the function is given as
f ( x ) = 6356 x + 5095
Rewrite the equation of the function properly by removing the excess spaces
So, we have
f(x) = 6356x + 5095
The above equation is a linear equation
A linear equation is represented as
f(x) = mx + c
As a general rule;
The domain of a linear equation is all set of real numbers
This is the same for the range
i.e. the range of a linear equation is all set of real numbers
When the set of real numbers is represented as an interval notation, we have the following representation
(oo, oo)
Hence, the domain is (oo, oo)
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Possible question
let f ( x ) = 6356 x + 5095 . Use interval notation to represent the domain of the function.
Many answers are possible.
octavius wants to write the equation of a line perpendicular to y=-4x + 5 that passes through the point (8,-3). Describe the mistake octavius made and write the correct equation of the line.
The equation of line perpendicular to 4y = x-8 passing through (4,-1) is:
[tex]y = \frac{1}{4} x-5[/tex].
What is a equation of line?These lines are written in the form y = mx + b, where m is the slope and b is the y-intercept. We know from the question that our slope is 3 and our y-intercept is –5, so plugging these values in we get the equation of our line to be y = 3x – 5.
Given equation of line is:
y=-4x + 5
Let [tex]m_{1}[/tex] be the slope of given line
Then,
[tex]m_{1}[/tex] = -4
Let [tex]m_{2}[/tex] be the slope of line perpendicular to given line
As we know that product of slopes of two perpendicular lines is -1.
[tex]m_{1}*m_{2} = -1\\- 4*m_{2}=-1\\ m_{2} = \frac{1}{4}[/tex]
The slope intercept form of line is given by
[tex]y = m_{2}x+c[/tex]
[tex]y = \frac{1}{4} x+c[/tex]
to find the value of c, putting (4,-1) in equation
[tex]-3= \frac{1}{4} *8+c\\-3-2 = c\\c = -5[/tex]
Putting the value of c in the equation
[tex]y=\frac{1}{4} x-5[/tex]
Hence, The equation of line perpendicular to 4y = x-8 passing through (4,-1) is [tex]y=\frac{1}{4} x-5[/tex].
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what is the sum of -1 1/3 + 3/4
Here, we want to add two fractions
What we have to do here is to make the mixed fractin an improper one
To do this, we multiply the denominator by the standing number, and add to the numerator, then we place the value over the denominator
Thus, we have it that;
[tex]\begin{gathered} 1\frac{1}{3}\text{ = }\frac{4}{3} \\ -\frac{4}{3}+\frac{3}{4}\text{ = }\frac{-16+9}{12}=\text{ }\frac{-7}{12} \end{gathered}[/tex]Rewrite y + 1 = -2x – 3 in standard form
The algebraic expressions can be written as
[tex]a+b+c=0[/tex]The given expression is,
[tex]\begin{gathered} y+1=-2x-3 \\ -2x-y=1+3 \\ -2x-y=4 \\ -2x-y-4=0 \\ 2x+y+4=0 \end{gathered}[/tex]