1. In how many different ways can 5 people be seated at a round table?
2. Each block on the grid is 1 unit by 1 unit. We wish to walk from A to B via 7 unit path but we have to stay on the grid. How many different paths can we take?
3. How many different "words" can be made from the given word by re-arranging the letters?
a) VECTOR
B) TRUST
Answers
Assume the people A, B, C, D and E are sitting in a row.
since there are 5 positions, they can sit in 5*4*3*2*1=120 ways.
consider 1 certain sitting position, for example ABCDE, that is B has A to his right, C to his left. D has C to his right and E to his left. And so on.
the positions
ABCDE
BCDEA
CDEAB
DEABC
EABCD
while in a row are different positions, in a round table they are the same thing.
(read the letters starting from A, and when the letters finish, continue reading from the beginning of the row. They all read "ABCDE")
This means that any arrangement in a round table, is converted to 5 arrangements in a row.
So there are in total [tex] \frac{5!}{5}=4*3*2*1=24 [/tex] arrangements of 5 people around a table.
2. check picture.
Consider the case where A is at (2, 2) and B is at (5, 6)
A can move to B through 3 horizontal units to the right, we call them E (for East) and 4 vertical units up, which we call N (for north).
the total path is 7 units.
it can be done in several ways, for example:
ENNNENE (the red path shown in the figure)
NNEEENN (the black path shown in the figure)
In total there are
[tex] \frac{7!}{3!*4!}= \frac{7*6*5*4!}{3!*4!}= \frac{7*6*5}{3!}= 35 [/tex] paths.
Remark: 7! is the number of arrangements if the letters were different. We divide by 3! because of the 3 E's and 4! because of the 4 N's.
Another possibility could be A'(3, -5), B'(8,-3).
From A' to B' we go by a total of 5 E and 2 N, so there are
[tex] \frac{7!}{2!*5!}= \frac{7*6*5!}{2!*5!}= \frac{7*6}{2}= 42 [/tex] paths in total.
there is one more possibility
[tex] \frac{7!}{1!(6!)}=7 [/tex]
C.
a) consider the letters {V,E,C,T,O,R}
we can form 6*5*4*3*2*1=720 words, as the first digit can be any of the 6 letters, combined with 5 for the second letter and so on.
b) the difference with a is that we have 5 letters and we have 2 T.
we have in total 5*4*3*2*1=120 arrangements of these letters, if the 2 T's were considered as separate.
consider the arrangements:
[tex]T_1RUST_2[/tex] and [tex]T_2RUST_1[/tex].
We read bth as just TRUST, but the permutation formula 5*4*3*2*1 considers these as 2 different.
this is why we need to divide 120 by 2, to get the actual number of words that can be formed. So the number is 60.
Answers:
1) 24
2) 7, 35 or 42. it depends on the positions of A and B, check the solution
3) a-720, b-160
Related Questions
Find the GCF of the first two terms and the GCF of the last two terms of the polynomial. 5h^3+20h^2+4h+16
Answers
The GCF of the last two terms: 4
5h³ and 20h²
Both of these terms are divisible by 5. (constant)
Both of these terms also have at least 2 h's. (variable)
4h and 16
Both of these terms are divisible by 4.
Only one term has a variable, so that has to be left out.
what must be done to the expression x + 108 to make it equal x?
Answers
To make the expression x + 108 equal to x, subtract 108 from both sides of the equation.
Explanation:To make the expression x + 108 equal to x, we need to subtract 108 from both sides of the equation. This will give us:
x + 108 - 108 = x - 108
Simplifying the equation, we get:
x = x - 108
However, if we are not solving an equation and simply need to simplify the expression x + 108 to make it equal to x, then the answer is that nothing can be done. The expression cannot be simplified to make it equal to x without additional information or context.
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Give an example of a function that is integrable on the interval [-1,1], but not continuous on [-1,1]. explain.
Answers
how about f(x) = |x|/x
It jumps from -1 to +1 at x=0, yet it can be integrated on [-1,1]. The surface is 2.
Graph the hyperbola with equation quantity x plus four squared divided by sixteen minus the quantity of y plus three squared divided by twenty five = 1.
Answers
(x-h)²/a² - (y-k)²/b² = 1, where
(h,k) is the center
a is the semi-major axis
b is the semi-minor axis
The given equation is
(x+4)²/16 - (y+3)²/25 = 1
So, from the general form we can deduce that,
Center(-4,-3)
a = 4
b = 5
So, the first point we can plot is the centerpoint. Next, you draw the two intersecting lines. Their slopes are +/- b/a. Thus, it corresponds to +/- 5/4. Using this slope, we can find the equation of the two lines by using the slope and the center.
-3 = +5/4 (-4) + b ---> b= 2
-3 = -5/4 (-4) + b ---> b= -8
So, you plot the equations y=5/4x + 2 and y = -5/4 x -8 by assigning values of x and plotting them against y. Then, the vertex of the hyperbolas are 4 units from the center, denoted by the green dots. The hyperbola is shown in the next picture.
Round 846 to the nearest hundred___________ and 3756 to the nearest thousand ________
Answers
Which of the following is FALSE? The diagonals of a parallelogram bisect each other. The diagonals of a rhombus bisect each other. The diagonals of a square bisect each other. The diagonals of a kite bisect each other
Answers
Answer:
The diagonals of a kite bisect each other
Step-by-step explanation:
Let analyse all possible answer:
a. The diagonals of a parallelogram bisect each other
True, it is one of the properties of a parallelogram
b. The diagonals of a rhombus bisect each other
True, In any rhombus, the diagonals (lines linking opposite corners) bisect each other at right angles (90°). That is, each diagonal cuts the other into two equal parts, and the angle where they cross is always 90 degrees.
c. The diagonals of a square bisect each other.
True, it is one of the properties of a square
d. The diagonals of a kite bisect each other
Wrong, because we do not know what shape of the kite is, it can have a really strange shape that we can not identify where the diagonals are as you can see in the attached photo.
Flying against the wind, an airplane travels 2280 km in 4 hours. Flying with the wind, the same plane travels 6650 km in 7 hours. What is the rate of the plane in still air and what is the rate of the wind?
Answers
let plane speed in still air is X and air speed Y
so, X-Y = 2280/4 = 570 km/h
and X+Y = 6650/7 = 950 km/h
so, X = (570 + 950) / 2 = 760 km/h
and Y = 950- 760 = 190 km/h
:)
What is the difference between discrete vs continuous data
And can someone please help me with 1-5
Answers
On the other hand, discrete data can't be treated the same way (eg: if n = number of people, then there is no midpoint between n = 3 and n = 4).
-------------------------------------
With that in mind, we have the following answers
1) Continuous data. Time values are always continuous. Any two distinct time values can be averaged to find the midpoint
2) Continuous data. Like time values, temperatures can be averaged as well.
3) Discrete data. Place locations in a race or competition are finite and we can't have midpoints. We can't have a midpoint between 9th and 10th place for instance.
4) Continuous data. We can find the midpoint and it makes sense to do so when it comes to speeds.
5) Discrete data. This is a finite number and countable. We cannot have 20.5 freshman for instance.
Find the volume of a cone with radius 30 in and slant height 50 in.
Answers
The volume of cone is 37,680 cubic inch.
What is volume?Space is used by every three-dimensional object. This space is estimated concerning its volume. The volume of an object in three-dimensional space is the amount of space it occupies within its boundaries. The capacity of the object is another name for it.
An object's volume can help us figure out how much water is needed to fill it, like how much water is needed to fill a bottle, aquarium, or water tank.
Given radius of cone = r = 30 inch
slant height = l = 50 inch
volume of cone = V = 1/3πr²h
relation between height, radius, and slant height is given by
l² = h² + r²
where l = slant height, h = height and r = radius
h = [tex]\sqrt{l^{2} - r^{2} }[/tex]
h = [tex]\sqrt{50^{2} - 30^{2} }[/tex]
h = √1600 = 40 inch
V = 1/3πr²h = 1/3π × (30)² × 40
V = (3.14 × 900 × 40)/3
V = 37,680 cubic inch
Hence the volume is 37,680 cubic inch.
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How can you rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations?
Answers
Make a the subject of the formula
b^2 = 2as + c
subtract c from both sides:-
b^2 - c = 2as
Now divide both sides by 2s ( to isolate a):-
a = (b^2 - c) / 2s
You can rearrange formulas to highlight a specific quantity of interest by identifying it and isolating it using inverse operations.
Explanation:In mathematics, you can rearrange formulas to highlight a specific quantity of interest using the same reasoning as in solving equations. The goal is to isolate the variable or the quantity you want to highlight on one side of the formula. Here are the steps to do it:
Identify the quantity of interest you want to highlight.
Apply inverse operations to isolate that quantity on one side of the equation/formula.
Simplify or solve for the highlighted quantity if needed.
For example, if you have the formula for the area of a rectangle, A = L * W, and you want to solve for the length of the rectangle, you can rearrange the formula as L = A / W, highlighting the length as the quantity of interest.
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NEED HELP!!!!!!!!!!!!!!!
Which factorizations can be used to identify the real zeros of the function f(x)=-20x^2+23x-6 ?
A. (-10x+2)(2x+3)
B. -(10x+2)(2x-3)
C. -(4x-3)(5x+2)
D. -(4x-3)(5x-2)
Answers
Answer:
Option D. [tex]-(4x-3)(5x-2)[/tex]
Step-by-step explanation:
we have
[tex]f(x)=-20x^{2}+23x-6[/tex]
Equate the function to zero
[tex]-20x^{2}+23x-6=0[/tex]
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]-20x^{2}+23x=6[/tex]
Factor the leading coefficient
[tex]-20(x^{2}-(23/20)x)=6[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]-20(x^{2}-(23/20)x+(529/1,600))=6-(529/80)[/tex]
[tex]-20(x^{2}-(23/20)x+(529/1,600))=-(49/80)[/tex]
[tex](x^{2}-(23/20)x+(529/1,600))=(49/1,600)[/tex]
Rewrite as perfect squares
[tex](x-(23/40))^{2}=(49/1,600)[/tex]
[tex](x-(23/40))=(+/-)(7/40)[/tex]
[tex]x=(23/40)(+/-)(7/40)[/tex]
[tex]x=(23/40)(+)(7/40)=30/40=3/4[/tex]
[tex]x=(23/40)(-)(7/40)=16/40=2/5[/tex]
therefore
[tex]-20x^{2}+23x-6=-20(x-(3/4))(x-(2/5))[/tex]
[tex]-20x^{2}+23x-6=-(5)(4)(x-(3/4))(x-(2/5))[/tex]
[tex]-20x^{2}+23x-6=-(4x-3)(5x-2)[/tex]
Melissa has three different positive integers. she adds their reciprocals together and gets a sum of 1. what is the product of her integers?
Answers
First let us assign the three positive integers to be x, y, and z.
From the given problem statement, we know that:
(1/x) + (1/y) + (1/z) = 1
Without loss of generality we can assume x < y
< z.
We know that:
1 = (1/3) + (1/3) + (1/3)
Where x = y = z = 3 would be a solution
However this could not be true because x, y, and z must all be different integers. And x, y, and z cannot all be 3 or bigger than 3 because the sum would then be less than 1. So let us say that x is a denominator that is less than 3. So x = 2, and we have:
(1/2) + (1/y) + (1/z) = 1
Therefore
(1/y) + (1/z) = 1/2
We also know that:
(1/4) + (1/4) = (1/2)
and y = z = 4 would be a solution, however this is also not true because y and z must also be different. And y and z cannot be larger than 4, so y=3, therefore
(1/2) + (1/3) + (1/z) = 1
Now we are left by 1 variable so we calculate for z. Multiply both sides by 6z:
3z + 2z + 6 = 6z
z = 6
Therefore:
(1/2) + (1/3) + (1/6) = 1
so {x,y,z}={2,3,6}
Final answer:
The product of Melissa's integers is 120.
Explanation:
The product of Melissa's integers is 120.
Let the integers be a, b, and c.
According to the given information, 1/a + 1/b + 1/c = 1.
By finding common denominators and simplifying, you can determine that a * b * c = 120.
Can you use matrix multiplication with a matrix with three rows and a matrix with two columns
Answers
What is the length of the diagonal of a cube with a side length of 5 cm? Round to the nearest tenth. 7.1 cm 8.7 cm 21.3 cm 26.7 cm
Answers
D = 5√3 ≈ 8.7 cm.
The required length of the diagonal of a cube with a side length of 5 cm is approximately 8.7 cm.
What is diagonal?A diagonal is a straight line that connects two non-adjacent corners of a polygon, such as a square, rectangle, or any other shape with four or more sides. For a given quesiton, the diagonal of a cube with side length s is given by the formula d = s√3.
Here,
As mentioned in the question, we have given a cube with a length of its side is 5 cm.
We know that the diagonal of a cube with side length s is given by the formula d = s√3.
Substituting s=5 into this formula, we get:
d = s√3.
d = 5√3
Rounding to the nearest tenth, we get d = 8.7 cm.
Therefore, the length of the diagonal of a cube with a side length of 5 cm is approximately 8.7 cm.
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How many days are between march the 15th and august the 15th?
Answers
August the 15 is 15 days into the month.
You add 16+15 which equals 31 days.
Answer:21 weeks and 6 days
Step-by-step explanation:
lol im late its 2021
A candy factory made 54 boxes of chocolates. Each box weighs 2 pounds. They packed the boxes in 6 cases with the same number of boxes in each case. How many pounds of chocolates were there in each case?
Answers
The number of pounds of chocolates in each case is 18 pounds
How to find the number of pounds in each case?
The candy factory made 54 boxes of chocolates.
Each box weighs 2 pounds
Therefore,
each box = 2 pounds
They packed the boxes in 6 cases with the same number of boxes in each case. Therefore,
Boxes in each case = 54 / 6 = 9 boxes
Therefore,
pounds of chocolates in each case = 9 × 2 = 18 pounds
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A fast typist can type 120 words in 100 seconds. In 180 seconds, how many words could be typed?
Answers
180 seconds --- x words
[tex]x= \cfrac{180*120}{100}=18*12=216 \ \text{words}[/tex]
Answer:
216
Step-by-step explanation:
i think
If a baseball team's ratio of wins to losses is 5 to 4, what is the ratio of wins to total games played?
Answers
Triangle LMN has vertexes at L(-1, -6), M(1, -6), and N(1, 1). Find the measure of angle L to the nearest degree.
Answers
is a right-triangle, and you can pretty much see how long "y" and "x" are.
make sure your calculator is in Degree mode.
The midpoint of A (-4, 2) and B(8, 5) is
Answers
Answer: The midpoint of the points A(-4, 2) and B(8, 5) is M(2, 3.5).
Step-by-step explanation: We are given to find the midpoint of the points A(-4, 2) and B(8, 5).
We know that
the co-ordinates of the midpoint of the points (a, b) and (c, d) are given by
[tex]M=\left(\dfrac{a+c}{2},\dfrac{b+d}{2}\right).[/tex]
Therefore, the co-ordinates of the midpoint of the points (-4, 2) and B(8, 5) will be
[tex]M=\left(\dfrac{-4+8}{2},\dfrac{2+5}{2}\right)=\left(\dfrac{4}{2},\dfrac{7}{2}\right)=(2,3.5).[/tex]
Thus, the midpoint of the points A(-4, 2) and B(8, 5) is M(2, 3.5).
What is the 12th term of the arithmetic sequence where a1 = 5 and a6 = 20?
Answers
an=nth term
a1=first term
d=common differnce
n=which term
alrighty, a1=5
neat
an=5+d(n-1)
given a6=20
20=a6=5+d(6-1)
20=5+d(5)
20=5+5d
minus 5 both sides
15=5d
divide both sides by 5
3=d
so
an=5+3(n-1) is the formula for the nth term
12th term
a12=5+3(12-1)
a12=5+3(11)
a12=5+33
a12=38
the 12th term is 38
How do you find the vertex of this parabola? I just plotted the points down on a graphing calculator, but I figure there's a better way.
Answers
Jonathan deposits $2000 into a bank account that pays 7% annual interest compounded annually. This means the bank pays him 7% of his account balance as interest at the end of each year, and he leaves the original amount and the interest in the account.
1. Write a recursive formula to model the situation described above. Use A0 to represent the initial amount invested. Then describe An, the next year’s balance, in terms of the current year’s balance, An-1.
Answers
A=p(1+r/100)^n
where;
A=future amount=An
p=current amount=A0
r=rate=7%
n=time= 1 year
thus rewriting our formula we shall have:
An=A0(1+r)^n
The above is the model describing the information given for future cash flow;
hence, the amount in 1 year will be:
An=A0(1+0.07)^1
An=2000(1.07)^1
An=$2,140
What does "a"equal to?
Answers
7.4a+22.2=3.2a+43.2
Subtract 3.2a from both sides
4.2a+22.2=43.2
Subtract 22.2 from both sides
4.2a=21
Divide both sides by 4.2
a=5
The table below shows the surface area y, in square feet, of a shrinking lake in x days: Time (x) (days) 5 10 15 20 Surface area (y) (square feet) 90 85 70 61 Part A: What is the most likely value of the correlation coefficient of the data in the table? Based on the correlation coefficient, describe the relationship between time and surface area of the lake. [Choose the value of the correlation coefficient from −1, −0.98, −0.5, −0.02.] (4 points) Part B: What is the value of the slope of the graph of surface area versus time between 15 and 20 days, and what does the slope represent? (3 points) Part C: Does the data in the table represent correlation or causation? Explain your answer. (3 points)
Answers
Given the table below showing the surface area y, in square feet, of a shrinking lake in x days.
Time (x) (hours): 5 10 15 20
Surface area (y) (square inches): 90 85 70 61
We can find the correlation coeficient of the data using the table below:
[tex]\begin{center} \begin{tabular} {|c|c|c|c|c|} x & y & x^2 & y^2 & xy \\ [1ex] 5 & 90 & 25 & 8,100 & 450\\ 10 & 85 & 100 & 7,225 & 850\\ 15 & 70 & 225 & 4,900 & 1,050\\ 20 & 61 & 400 & 3,721 & 1,220\\ [1ex] \Sigma x=50 & \Sigma y=306 & \Sigma x^2=750 & \Sigma y^2=23,946 & \Sigma xy=3,570 \end{tabular} \end{center}[/tex]
Recall that the correlation coefitient is given by the equation:
[tex]r= \frac{n(\Sigma xy)-(\Sigma x)(\Sigma y)}{ \sqrt{(n\Sigma x^2-(\Sigma x)^2)(n\Sigma y^2-(\Sigma y)^2)} } \\ \\ = \frac{4(3,570)-(50)(306)}{ \sqrt{(4(750)-(50)^2)(4(23,946)-(306)^2)} } \\ \\ = \frac{14,280-15,300}{ \sqrt{(3,000-2,500)(95,784-93,636)}} = \frac{-1,020}{ \sqrt{500(2,148)}} \\ \\ = \frac{-1,020}{ \sqrt{1,074,000} } = \frac{-1,020}{1,036} =-0.98[/tex]
From the value of the correlation coeffeicient, it can be deduced that the surface area of the lake has a strong negative relationship with the time.
Recall the for the value of the correlation coeficient closer to +1, the relationship is strong positive, for the value closer to -1, the value is strong negative and for the values closer to zero, either way of zero is a weak positive if it is positive and weak negative if it is negative.
PART B:
Recall that the slope of a straight line passing through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]
is given by
[tex]m= \frac{y_2-y_1}{x_2-x_1}[/tex]
Thus, the slope of the graph of surface area versus time between 15 and 20 days, [i.e. the line passes through points (15, 70) and (20, 61)] is given by
[tex]m= \frac{61-70}{20-15}= \frac{-9}{5} =-1.8[/tex]
The value of the slope means that the surface area y, in square feet, of the shrinking lake, shrinks by 1.8 square feet every day between the day 15 and day 20.
PART C:
We can say that the data above represent both correlation and causation.
Recall that correlation expresses the relationship between two variables while causation expresses that an event is as a result of another event.
From the information above, we have seen that there is a relationship (correlation) between the passing of days and the shrink in the surface area of the lake.
Also we can conclude that the shrink in the surface area of the lake is a function of the passing of days, i.e. the shrink in the surface area of the lake is as a result of the passing of days.
Therefore, the data in the table represent both correlation and causation.
Which point is a solution of x + 2y ≤ 4?
A. (2,4)
B. (1,1)
C. (3,5)
D. (-1,5)
Answers
the distance across the center of a circle is called the?
Answers
The distance from one side of a circle to the other going through the center is called the Diameter.
I hope that helps!
There were 490 people at play. The admission price was $3.00 for adults and $1.00 for children. The admission receipts were $990. How many adults and children attended
Answers
3a + c = 990
3(490 - c) + c = 990
1470 - 3c + c = 990
-3c + c = 990 - 1470
-2c = - 480
c = -480 / -2
c = 240 <=== there were 240 children
a + c = 490
a + 240 = 490
a = 490 - 240
a = 250 <== there were 250 adults
Which regular polygon can be drawn by using rotations in geometry software?
Answers
Since we can draw a Hexagon because 360/6 = 60 degree.So we can draw a hexagon using 60 degree rotation(where result will be your exterior angle. So in this we can say 60 degree will be exterior angle.)
Similarly we can draw a 12 - gon. because 360/12 = 30 degree.So we can draw a 12 gon using 30 degree rotation.
Answer:
n- regular polygon
Step-by-step explanation:
In order to draw n- regular polygons, 360 must be divisible by n.
Therefore, there are many regular polygons such as 12- polygon using 30° rotations and 15-polygon using 24° rotations.
What is the domain of the relation: (0, 7), (8, -1), (2, 3), (-4, 6)?
A. (-4, 0, 2, 8)
B. (-1, 3, 6, 7)
C. (1, 7, 8, -1)
D. (2, 3, -4, 6)
Answers
When asked for the domain and given a set of ordered pairs, the domain is always all of the x values ordered from least to greatest.