Your integrand is missing some symbols. My best interpretation is the following integral:
[tex]I=\displaystyle\int\frac{x^4+18x^2+4}{x^5+30x^3+20x}\,\mathrm dx[/tex]
Decompose into partial fractions; we're looking for an expansion of the form
[tex]\dfrac{x^4+18x^2+4}{x^5+30x^3+20x}=\dfrac ax+\dfrac{bx^3+cx^2+dx+e}{x^4+30x^2+20}[/tex]
Now:
[tex]x^4+18x^2+4=a(x^4+30x^2+20)+(bx^3+cx^2+dx+e)x[/tex]
[tex]=(a+b)x^4+cx^3+(30a+d)x^2+ex+20a[/tex]
Matching up coefficients tells us that
[tex]\begin{cases}a+b=1\\c=0\\30a+d=18\\e=0\\20a=4\end{cases}\implies a=\dfrac15,b=\dfrac45,d=12[/tex]
so that
[tex]I=\displaystyle\frac15\int\frac{\mathrm dx}x+\frac45\int\frac{x^3+15x}{x^4+30x^2+20}\,\mathrm dx[/tex]
The integral is trivial:
[tex]\displaystyle\frac15\int\frac{\mathrm dx}x=\frac15\ln|x|+C[/tex]
For the second integral, notice that
[tex]\mathrm d(x^4+30x^2+20)=(4x^3+60x)\,\mathrm dx[/tex]
Distribute the 4 over the numerator, then substitute [tex]u=x^4+30x^2+20[/tex] and [tex]\mathrm du=(4x^3+60x)\,\mathrm dx[/tex]:
[tex]\displaystyle\frac15\int\frac{4x^3+60x}{x^4+30x^2+20}\,\mathrm dx=\frac15\int\frac{\mathrm du}u=\frac15\ln|u|+C=\frac15\ln(x^4+30x^2+20)+C[/tex]
So we have
[tex]I=\dfrac15\ln|x|+\dfrac15\ln(x^4+30x^2+20)+C[/tex]
and with some simplification,
[tex]I=\boxed{\ln\sqrt[5]{|x^5+30x^3+20x|}+C}[/tex]
What does volume measure? the amount of space occupied by a two-dimensional solid object the total area of all the surfaces of a three-dimensional solid object the amount of space inside the boundary of a two-dimensional object the amount of space occupied by a three-dimensional solid object
Answer:
the amount of space occupied by a three-dimensional solid object
Step-by-step explanation:
Volume is a measure of the space in a 3D solid object enclosed by the closed surfaces of the solid object.
By using the definition of volume, we can see that the correct option is the last one:
"The amount of space occupied by a three-dimensional solid object"
What does volume measure?
Volume is defined as a 3-dimensional metric derived from longitude, that measures a region in the space. So, each region that "takes space" has a volume.
With that in mind, the option that correctly describes volume is the last option:
"The amount of space occupied by a three-dimensional solid object"
If you want to learn more about volume, you can read:
https://brainly.com/question/1972490
HELP PLEASEEE!!!!!!!!!!
Answer:
100
Step-by-step explanation:
height = constant/ width
Taking the point (5,20)
where 5 is the width and 20 is the height
20 = constant/ 5
Multiply each side by 5
5*20 = constant
100 = constant
Identify the algebraic series.
A) 10, 23, 36...... B) 4 + 8 + 16 +...... C) 100, 90, 81,...... D) 84 + 73 + 62 +.......
Answer:
Please read the answer below.
each number is the previous one less 11.
Step-by-step explanation:
A)
10, 23, 36, 36 +13 , 36 +13 +13 , 36 +13 +13+13, ...
= 10, 23, 36, 49, 62, 75, ...
each number is the sum of the previous number plus 13.
B)
4 + 8 + 16 + 2^5 + 2^6 +2^7 +2^8 + ... + 2^n
= 4 + 8 +16 + 32 + 64 + 128 +256 ...
each number is a power of 2.
C)
100, 90, 81, 81-9, 81-9-9, 81-9-9-9, ...
= 100, 90, 81, 72, 63, 54, 45, ...
each number is the previous one less 9.
D)
84 + 73 + 62 + (62-11) + (62-11-11) + (62-11-11-11) + ....
= 84 + 73 +62 + 51 + 40 + 39 + ...
each number is the previous one less 11.
Using traditional methods, it takes 9.5 hours to receive a basic flying license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique with 15 students and observed that they had a mean of 10.0 hours with a standard deviation of 1.6. A level of significance of 0.1 will be used to determine if the technique performs differently than the traditional method. Assume the population distribution is approximately normal. Make the decision to reject or fail to reject the null hypothesis.
Answer:
Step-by-step explanation:
Null hypothesis: u = 9.5hrs
Alternative: u =/ 9.5hrs
Using the t test
t = x-u/sd/√n
Where x is 10hrs, u is 9.5, sd is 1.6 and n is 15
t = 10-9.5 / (1.6/√15)
t = 0.5 / (0.4131)
t = 1.21
In order to make a conclusion, we have to find the p value at a significance level lot 0.1. The p value is 0.2263 which is greater than 0.1. This, we will fail to reject the null hypothesis and conclude that there is not enough statistical evidence to prove that the technique performs differently than the traditional method.
I need help, I don’t need an explanation, just the answer
Answer:
[tex]x = 48 \: \: \: \: \: \: y = 16[/tex]
Step-by-step explanation:
[tex]y = - x + 4y[/tex]
[tex]y + x = 4y[/tex]
[tex]y + x - 4y = 0[/tex]
[tex] - 3y + x = 0[/tex]
[tex] - x + 4y = 16[/tex]
[tex] - 3y + x = 0[/tex]
[tex] - 3y = - x[/tex]
[tex]y = - \frac{1}{3} ( - 1)x[/tex]
[tex]y = \frac{1}{3} x[/tex]
[tex]4 \times ( \frac{1}{3} )x - x = 16[/tex]
[tex] \frac{1}{3}x = 16 [/tex]
[tex]x = 48[/tex]
[tex]y = \frac{1}{3} \times 48[/tex]
[tex]y = 16[/tex]
Hope this is correct and helpful
HAVE A GOOD DAY!
Answer:
x = 48, y = 16
Step-by-step explanation:
11. Caroline wraps packages at a store. She wraps
9 packages each hour. Which statement is true
about the number of packages she wraps?
A. In 2 hours, Caroline wraps an odd number of
packages.
B. In 3 hours, Caroline wraps an even number of
packages.
C. In 5 hours, Caroline wraps an odd number of
packages.
D. In 7 hours, Caroline wraps an even number of
packages.
Answer:
C. in five hours Caroline wraps an odd number of packages
Step-by-step explanation:
for A until hours you would multiply 2 by 9 and 2 by 9 is 18 and that's an even number so it's not A.
A eliminated.
for B in 3 hours 3 by 9 is 27 and that's an odd number so B is automatically eliminated.
for C in 5 hours all you would do is multiply the 9 by 5 and 9 by 5 is 45 and 45 is indeed an odd number so C is your answer.
for D 7 by 9 is 63 and 63 is an odd number so we already know that C is the answer but still we got to check and D is wrong because 63 is not an even number.
Find the directional derivative of f at the given point in the direction indicated by the angle θ. f(x, y) = y cos(xy), (0, 1), θ = π/3
Answer:
√3/2
Explanation:
The directional derivative at the given point is gotten using the formula;
∇f(x,y)•u where u is the unit vector in that direction.
∇f(x,y) = f/x i + f/y j
Given the function f(x, y) = y cos(xy),
f/x = -y²sin(xy) and
f/y = -xysin(xy)+cos(xy)
∇f(x,y) = -y²sin(xy) i + (cos(xy)-xysin(xy)) j
∇f(x,y) at (0,1) will give;
∇f(0,1) = -0sin0 i + cos0j
∇f(0,1) = 0i+j
The unit vector in the direction of angle θ is given as u = cosθ i + sinθ j
u = cos(π/3)i+ sin(π/3)j
u = 1/2 i + √3/2 j
Taking the dot product of both vectors;
∇f(x,y)•u = (0i+j)•(1/2 i + √3/2 j)
Note that i.i = j.j = 1 and i.j = 0
∇f(x,y)•u = 0 + √3/2
∇f(x,y)•u = √3/2
The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{\sqrt{3}}{2}[/tex].
How to calculate the directional derivative of a multivariate functionThe directional derivative is represented by the following formula:
[tex]\nabla_{\vec v} f = \nabla f(x_{o},y_{o}) \cdot \vec v[/tex] (1)
Where:
[tex]\nabla f(x_{o}, y_{o})[/tex] - Gradient evaluated at point [tex](x_{o},y_{o})[/tex].[tex]\vec v[/tex] - Directional vectorThe gradient of [tex]f[/tex] is calculated below:
[tex]\nabla f (x_{o},y_{o}) = \left[\begin{array}{cc}\frac{\partial f}{\partial x} (x_{o}, y_{o}) \\\frac{\partial f}{\partial y} (x_{o}, y_{o})\end{array}\right][/tex] (2)
Where [tex]\frac{\partial f}{\partial x}[/tex] and [tex]\frac{\partial f}{\partial y}[/tex] are the partial derivatives with respect to [tex]x[/tex] and [tex]y[/tex], respectively.
If we know that [tex](x_{o}, y_{o}) = (0, 1)[/tex], then the gradient is:
[tex]\nabla f(x_{o}, y_{o}) = \left[\begin{array}{cc}-y^{2}\cdot \sin xy\\\cos xy -x\cdot y\cdot \sin xy\end{array}\right][/tex]
[tex]\nabla f (x_{o}, y_{o}) = \left[\begin{array}{cc}-1^{2}\cdot \sin 0\\\cos 0-0\cdot 1\cdot \sin 0\end{array}\right][/tex]
[tex]\nabla f (x_{o}, y_{o}) = \left[\begin{array}{cc}0\\1\end{array}\right][/tex]
If we know that [tex]\vec v = \cos \frac{\pi}{3}\,\hat{i} + \sin \frac{\pi}{3} \,\hat{j}[/tex], then the directional derivative is:
[tex]\Delta_{\vec v} f = \left[\begin{array}{cc}0\\1\end{array}\right]\cdot \left[\begin{array}{cc}\cos \frac{\pi}{3} \\\sin \frac{\pi}{3} \end{array}\right][/tex]
[tex]\nabla_{\vec v} f = (0)\cdot \cos \frac{\pi}{3} + (1)\cdot \sin \frac{\pi}{3}[/tex]
[tex]\nabla_{\vec v} f = \frac{\sqrt{3}}{2}[/tex]
The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{\sqrt{3}}{2}[/tex]. [tex]\blacksquare[/tex]
To learn more on directional derivatives, we kindly invite to check this verified question: https://brainly.com/question/9964491
do following division with polynomials
1) (x^3-2x^2+3x-3)÷(x+2)
The energy transfer diagram represents the energy of a person diving into a pool. A wide arrow labeled gravitational potential energy 8000 J with a small arrow turning away from it labeled thermal energy ? J. The rest of the arrow goes forward labeled kinetic energy 7000 J. How much thermal energy is generated? 1000 J 2000 J 7000 J 15,000 J
Answer:
The amount of thermal energy that is generated is 1000 joules.
Step-by-step explanation:
Let consider the person diving into a pool as a closed system, that is, a system with only energy interaction with the surroundings. According to the First Law of Thermodynamics, which is a generalization of the Principle of Energy Conservation, the total energy of the system is neither destroyed, nor created, only transformed. The initial gravitational potential energy is transformed into translational kinetic and thermal energy. That is to say:
[tex]U_{g} - K - Q = 0[/tex]
Where:
[tex]U_{g}[/tex] - Gravitational potential energy, measured in joules.
[tex]K_{tr}[/tex] - Translational kinetic energy, measured in joules.
[tex]Q[/tex] - Thermal energy, measured in joules.
The thermal energy is now cleared:
[tex]Q = U_{g}-K[/tex]
Given that [tex]U_{g} = 8000\,J[/tex] and [tex]K = 7000\,J[/tex], the amount of thermal energy that is generated is:
[tex]Q = 8000\,J - 7000\,J[/tex]
[tex]Q = 1000\,J[/tex]
The amount of thermal energy that is generated is 1000 joules.
Answer:
1000J
Step-by-step explanation:
EDGE2020
A new cola company is testing to see what proportion of their cans contain at least 12 oz. If they want to be within 3% of the actual percentage, how many cans should they measure to be 90% confident
Answer: 752
Step-by-step explanation:
Given that,
Margin of error = 3% = 0.03
confidence level = 90% = 0.90
therefore from the z-table
z = 1.645
Now since no prior estimate of p is given, so we say p = 0.5
Sample size required will be
n = 1.645² × 0.5 ×(1-0.5) / 0.03² = 751.67
n = 751.67 ≈ 752
What is the greatest whole number that must be a divisor of the product of any three consecutive positive integers?
Answer:
6
Step-by-step explanation:
We can establish 6 as an upper bound, since 1*2*3 = 6, and 6 is clearly the greatest number that is a divisor of itself.
We can show that the product of any three consecutive numbers is divisible by 6, because out of any three consecutive integers, at least one must be divisible by 3, and at least one must be divisible by 2. Since the product must have factors of 2 and 3, it must also have 6 as a factor.
2x + y = 15
x = 15 - 2y
is it no solution?
Answer:
It has a solution.
Step-by-step explanation:
Step 1: Rearrange 1st equation into slope-intercept form
2x + y = 15
y = 15 - 2x
Step 2: Rearrange 2nd equation into slope-intercept form
x = 15 - 2y
2y + x = 15
2y = 15 - x
y = 15/2 - x/2
Step 3: Rewrite systems of equations
y = 15 - 2x
y = 15/2 - x/2
Since the two lines are not parallel, they will have a solution.
Strontium 90 is a radioactive material that decays according to the function Upper A (t )equals Upper A 0 e Superscript negative 0.0244 t Baseline commaA(t)=A0e−0.0244t, where Upper A 0A0 is the initial amount present and A is the amount present at time t (in years). Assume that a scientist has a sample of 500500 grams of strontium 90. (a) What is the decay rate of strontium 90?
Answer:
Decay rate K = -2.44%
Step-by-step explanation:
From the question, we want to know the decay rate of strontium 90
Mathematically, this is accessible from its decay equation
From the decay equation, we can see that that ;
At = Ao e^-0.0244t
Generally, the decay equation of a radioactive sample can be written as
At = Ao e^-kt
where K represents the decay constant
From the equation, we can see that;
k = 0.0244 which when represented as a percentage is 2.44%
Since it’s a decay we can say that the decay rate is -2.44%
The answer is : -2.44%
What is the least number of colors you need to correctly color in the sections of the pictures so that no two touching sections are the same color?
Answer:
8 colors
Step-by-step explanation:
There should be at least 8 different colors available for coloring the sections. The one color is used to color all the small triangles on the upper most and lower most lines, then there will be required another color so that the edges does not matches with the previous color. For the bigger hexagon shapes in the center we will require different colors for all of them because all of the hexagon shapes touches a line and an edge with each other.
Answer:
its 2 trust me
Step-by-step explanation:
its two cause if you think about it and color in the hexagons and triangles two different colors it works
logx - logx - 1^2 = x - 4
Answer:
Look at the image below↓
The Digit 6 in which numbers repesent a value of 6 ones?
Answer:
B. 16.4
Step-by-step explanation:
Well in 16.4 the digit 6 is the first number meaning it is 6 ones.
Thus,
choice b is correct.
Hope this helps :)
Could someone please help. The given sides are 25 at the left, 35 at bottoms, 5 at the right. Thank you :)
Answer:
120 cm
Step-by-step explanation:
If you rearrange the lines you will get a rectangle whose sides are 25 cm and 35 cm.
To get a triangle, move the lines inside the rectangle keeping them parallel to the lines in front of them.
Let P the perimeter of the rectangle:
P = 25*2 + 35*2 = 50+70 = 120 cm
Find the slope of the line that passes through the following points (show work) E(4, 1 2/3) and F(-2, 2/3)
Answer:
1/6
Step-by-step explanation:
The formula of a slope [tex]\frac{y2-y1}{x2-x1}[/tex] (change in y / change in x)
1. Substitute in the values
(x1, y1) = (4, 1[tex]\frac{2}{3}[/tex])
(x2, y2) = (-2, [tex]\frac{2}{3}[/tex])
([tex]\frac{2}{3}[/tex] - 1[tex]\frac{2}{3}[/tex]) / (-2 - 4)
2. Solve
[tex]\frac{2}{3}[/tex] - 1[tex]\frac{2}{3}[/tex] = -1
-2 - 4 = -6
slope = [tex]\frac{-1}{-6}[/tex] = [tex]\frac{1}{6}[/tex]
NEED HELP THANKLSSSS
Answer:
Side length: 3 cm.
Surface area: 54 cm squared.
Step-by-step explanation:
The formula for a cube is the side length cubed, since the formula for a rectangular prism is length times width times height. Those three measurements are the same for a cube.
So, since the volume is 27 cm cubed, we can say that the side length of the cube is the cube root of 27 cm cubed, or 3 cm.
There are 6 sides on a cube, and every cube has the same area. Since the side length of the cube is 3 cm, the area of one side of the cube is 3 * 3 = 9 cm squared. 9 * 6 = 54 cm squared.
Hope this helps!
Lana has a friend who is deaf and uses sign language to communicate. She can
sign about 36 concepts per minute. How many concepts can she sign in a 15
minute period?
Information given:
1. 36 concepts per minute
2. The question is telling us how many concepts can she sign in 15 min period.
_________________________________________
So we know that we have to multiply to find the answer.
36x15 will give us 540 as the answer.
So the answer to your question is 540
I attached a picture for you to see how I had multiply.
Hope this helps! :)
Simplify the following algebraic expression.
square root of 392x^7
Answer:
[tex] \sqrt{392 {x}^{7} } [/tex]
Simplify
that's
[tex] \sqrt{392} \times \sqrt{ {x}^{7} } \\ \\ = \sqrt{196 \times 2} \: \times \sqrt{ {x}^{7} } \\ \\ = 14 \sqrt{2} \times \sqrt{ {x}^{7} } \\ \\ = 14 \sqrt{2x ^{7} } [/tex]
Hope this helps you
5x + y = 6
5x + 3y = -4
The y-coordinate of the solution to the system shown is
-5
-1
1/2
1
Answer:
y coordinate of the solution to the system is -5
Answer:
-5
Step-by-step explanation:
n
The nth term of a sequence is given by
T = -19n - 3.
(a) Which term of the sequence has a value
of -250?
b) Is-344 a term in the sequence? Why?
Answer:
a)13 b)no because at 18th term its -345
Three of the sides will require fencing and the fourth wall already exists. If the farmer has 116 feet of fencing, what are the dimensions of the region with the largest area
Answer:
29 ft x 58 ft
Step-by-step explanation:
Let x be the length of each side perpendicular to the wall, and y be the length of the side parallel to the wall.
The amount of wire available is:
[tex]116 = 2x+y\\y=116-2x[/tex]
The area of the region is:
[tex]A=xy=x(116-2x)\\A(x)=116x-2x^2[/tex]
The value of 'x' for which the derivate of the area function is zero will yield the maximum area:
[tex]A(x)=116x-2x^2\\A'(x) = 116-4x=0\\x=29\ ft[/tex]
The value of y is:
[tex]y=116-2*29\\y=58\ ft[/tex]
The dimensions of the region with the largest area are 29 ft x 58 ft.
How many even 3 digit positive integers can be written using the numbers 3,4,5,6,and 7?
Answer:
I got 45, but I may be wrong.
Step-by-step explanation:
When a number is even, the number must end in an even number. Here, the even numbers are 4 and 6, so the numbers we are going to create are all going to end in 4 and 6.
To answer this question, we just have to find as many possible combinations following the guidelines provided.
334
344
354
364
374
434
444
454
464
474
534
544
554
564
574
634
644
654
664
674
336
346
356
366
376
436
446
456
466
476
536
546
556
566
576
636
646
656
666
676
736
746
756
766
776
Factor.
-7x2 + 49x
-7(x - 7)
01 - 7x2 + 49x)
-x(x – 7)
-7x(x – 7)
Answer:
Option 1: -7x (x - 7)
Step-by-step explanation:
Factor -7x out of -7x^2.
= -7x (x) + 49x
Factor -7x out of 49x.
= -7x (x) -7x (-7)
Factor -7x out of -7x (x) - 7x (-7).
= -7x (x - 7)
(Also I took the test and got this answer right)
What is closest to the area of this section of the garden?
Answer:
117.984ft^2
Step-by-step explanation:
To find the area of a circle the formula is PiR^2 so the area for the full 360 degree circle is. 530.929 ft^2.
Now that we know this we can use a ratio.
530.929/360=x/80
u divide out the left side and multiply it by 80.
so that shows that the area of this circle is
117.984ft^2
Consider the line – 5x – 8y= 3.
What is the slope of a line perpendicular to this line?
What is the slope of a line parallel to this line?
Answer:
Perpendicular Slope: 8/5
Parallel Slope: -5/8
Step-by-step explanation:
First, let's rewrite the line into slope intercept form.
-5x - 8y = 3
-8y = 5x + 3
y = -5x/8 + -3/8
Okay, so now we know the slope, -5/8, and the y-intercept, -3/8.
For a line to be perpendicular, the slope needs to be opposite of the given line's slope. This will cause the two lines to cross at a 90-degree angle, and therefore be perpendicular.
So a perpendicular line could be as follows:
y = 8x/5 + -3/8
So the perpendicular slope would be 8/5.
For a line to be parallel, the slope needs to be the same so that the two lines will never cross.
So a parallel line could be as follows:
y = -5x/8 + 1
So the parallel slope would be -5/8.
Cheers.
Answer:
Perpendicular Slope: [tex]\boxed{\frac{8}{5}}[/tex]
Parallel Slope: [tex]\boxed{-\frac{5}{8}}[/tex]
Step-by-step explanation:
Part 1: Rewrite into slope-intercept form
Firstly, the equations are written in standard form and not slope-intercept form, so to change that, follow the steps below.
Note: Remember the slope-intercept form equation - [tex]\boxed{y=mx+b}[/tex]
[tex]-5x-8y=3\\\\5x + (-5x-8y)=3+5x\\\\-8y=5x+3\\\\\frac{-8y}{-8} =\frac{5x+3}{-8} \\[/tex]
[tex]y=-\frac{5}{8}x-\frac{3}{8}[/tex]
Add [tex]5x[/tex] to both sides of the equation to isolate the y-variable. Then, divide by the coefficient of y to isolate it entirely. The equation is now in slope-intercept form.
Part 2: Determine the perpendicular slope
Perpendicular slopes are reciprocals of the given slopes. To turn the original slope into its reciprocal counterpart, follow these steps:
If the current slope is positive, add a negative sign. If the current slope is negative, remove the negative sign.The denominator becomes the numerator and the numerator becomes the denominator.To follow this for the slope of the given equation:
[tex]\boxed{-\frac{5}{8} \dashrightarrow \frac{8}{5} }[/tex]Part 3: Determine the parallel slope
Parallel slopes are equal - otherwise, the lines would eventually intersect. Therefore, the given slope is also the parallel slope.
The parallel slope is [tex]\boxed{-\frac{5}{8}}[/tex].
Jahlil is 6 inches shorter than 4 times his sister’s height. Jahlil’s height is 70 inches. Which equation represents h, his sister’s height in inches?
Answer:
4h - 6 = 70.
Step-by-step explanation:
Jahill's height is 70 inches. That is 6 inches shorter than 4 times his sister's height, h. That is the same thing as 4 times h minus 6.
70 = 4 * h - 6
4 * h - 6 = 70
4h - 6 = 70
4h = 76
h = 19.
His sister is 19 inches tall.
Hope this helps!
Answer:
Hello! The answer will be below! :3
Step-by-step explanation:
Question: Jahlil is 6 inches shorter than 4 times his sister’s height. Jahlil’s height is 70 inches. Which equation represents h, his sister’s height in inches?
Answer:The answer is 4h - 6 =70
(His sister is 19 inches tall)
Steps:
4 * h - 6 = 70
So, 4h - 6 = 70......
And 4h will be 76 which means h will equal to 19
His sisters hight is about 19in. tall.....
Hope this helps! :)
⭐️Have a wonderful day!⭐️
1 - Fill the space blanks
If we make a sequence selecting three elements from three different elements
{1, 2, 3} and we permit overlapped elements for the sequence, then the total
number of sequences is [ ] . If we do not take into account the order, the total
number of the selections is [ ] .
I'm totally lost in this, what is overlapped elements? This is about what math content? And what is the answer? Please i need help.
Answer:
The first part is of permutations.
We are selecting 3 elements from three different elements {1,2,3}
Points given:
We permit overlapped elements for the sequence. Here "overlapped elements" indicates that repetition is allowed.
So when repetition is allowed and order matters, we use permutations.
Formula to compute permutation is:
Lets say n is the three elements {1,2,3}
We have to select 3 elements so r = 3
Total number of selections using permutations = [tex]n^{r}[/tex] = n × n × n
= 3³ = 3 * 3 * 3
= 27
This means if we have 3 different elements then we have have 3 choices each time for making a sequence.
Hence If we make a sequence selecting three elements from three different elements {1, 2, 3} and we permit overlapped elements for the sequence, then the total number of sequences is 27.
Step-by-step explanation:
The second part indicates combinations.
This is because the statement of the question: If we do not take into account the order.
When the order does not matter, we use combinations.
So when the order does not matter and repetition is allowed we use the following formula:
Total number of selections using combinations = (r + n - 1)! / r! (n - 1)!
= (3 + 3 - 1) ! / 3! (3 - 1)!
= (3 + 2) ! / 3! (2!)
= 5! / 3! 2!
= 5*4*3*2*1 / (3*2*1 ) (2*1)
= 120 / 6 * 2
= 120 / 12
= 10
So these are the number of combinations of 3 elements taken 3 at a time with repetition.
The total number that will be selected in the permutations is 27.
How to calculate the permutations?Based on the information given, the total number of permutations will be:
= n³
= 3 × 3 × 3
= 27
Also, the total number of selection using combination will be:
= (3 + 3 - 1)! / 3!(3 - 1)!
= 120 / (6 × 2)
= 120/12
= 10
Learn more about permutations on:
https://brainly.com/question/1216161