Combinatorics
If Connie had only 7 pairs of shoes and she has room for 7 pairs of shoes, then she has only one way to take her shoes.
If she had 8 pairs of shoes, then she can select any group of shoes that leaves one pair out. This makes 8 possible ways to choose.
When the number of pairs of shoes goes up, then the counting gets more complex. That is when combinatorics is a useful tool.
If we have a total of n elements to select m, where the order of selection is not important, then the total number of selections is given by:
[tex]C_{n,m}=\frac{n!}{(n-m)!\cdot m!}[/tex]Where the sign (!) is the factorial of a number.
Connie has n=16 pairs of shoes and she will take m=7 from them, thus the number of possible ways or combinations is:
[tex]\begin{gathered} C_{16,7}=\frac{16!}{(16-7)!\cdot7!} \\ C_{16,7}=\frac{16!}{(9)!\cdot7!} \end{gathered}[/tex]Expanding the factorial down to match the greatest factorial in the denominator:
[tex]C_{16,7}=\frac{16\cdot15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9!}{(9)!\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}[/tex]Simplifying and calculating:
[tex]C_{16,7}=\frac{57,657,600}{5,040}=11,440[/tex]Connie can choose in 11,440 ways the shoes to choose
Explain how you know and i'll mark as brainlist.
Answer: They have the same area.
Step-by-step explanation:
Here you just have to count the small squares.
In the blue shape, you can see there are 16 small squares.
And in the green shape, there is also 16 squares.
And the small squares are all equal.
So they both have the same area.
Choose the term that has the given definition.AngleCircleCollinearCoplanarCongruentLine segmentParallel linesPerpendicular lines
Solution
Circle: The set of all points in a plane that lie the same distance from a single point in the plane.
Perpendicular lines: Intersecting lines that form right angles
Congruent: Having exactly the same shape and size
Angle: A figure formed by two rays that have the same endpoint
Simplify the given expression below:(1 + 2i) ⋅ (5 − 3i) (2 points) a2 − 15i b3 + 2i c5 − 6i d11 + 7i
Simplification of Algebraic Expression.
[tex]\begin{gathered} (1+2i)(5-3i) \\ \text{First, we clear the brackets. } \\ 1(5-3i)\text{ +2i(5 - 3i)} \\ \text{Multiplying through the brackets, we get} \\ 5\text{ -}3i+10i-6(-1) \\ \text{Note: i}^2=\text{ -1} \\ 5\text{ -3i + 10i +6} \\ C\text{ollecting like terms, we get} \\ 5\text{ + 6 -3i + 10i} \\ 11\text{ +7i} \end{gathered}[/tex]Thus, the correct answer is 11 +7i ( option D )
3.Six Flags charge and entrance fee per person and a fee per every ride that you get on. When 6 teachers went to Six Flags and rode on 20 rides total, they ended up paying $350. When 15 people went to Six Flags and rode on 47 rides they paid $900. Write an expression for this situation.
Answer:
Step-by-step explanation:
Expression:
y = mx + n*b
In which m is the cost per ride and b is the entrance fee.
y is the total cost, x is the number of rides and n is the number of people.
When 6 teachers went to Six Flags and rode on 20 rides total, they ended up paying $350.
This means that:
20m + 6b = 350
When 15 people went to Six Flags and rode on 47 rides they paid $900.
This means that:
47m + 15b = 900
Solving the system:
20m + 6b = 350
47m + 15b = 900
Multiplying the first equation by -5, the second by 2.
-100m - 30b = -1750
94m + 30b = 1800
-6m = 50
Which economic system has no government involvement in the market? a. capitalism b. communism c. socialism d. No economic system is free from government involvement.
Answer Is D
The economic system that has no government involvement in the market is d. No economic system is free from government involvement.
Why there is no economic system is free from government involvement?In reality there is no government intervention because at some point the government will comes in , this could actually be mininmal, only in the free market is where there is no government intervention.
It should be noted that in this free market, there is an unregulated system of economic exchange, where the act of taxes collection as well as quality controls, has no economic interventions which can be attributed to government .
Therefore option D is correct.
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How do I do pi equations?
Given:
The objective is to explain the method of solving pi equations.
The term with pi can be solved
I am struggling on figuring out how to do this. Please help me
Given:-
[tex]f(x)=13x+2[/tex]To find the inverse of the function.
So now we change the value of x in the given function as y and find the value of y. so we get,
[tex]x=13y+2[/tex]Now we find the value of y. so we get,
[tex]\begin{gathered} 13y+2=x \\ 13y=x-2 \\ y=\frac{x-2}{13} \end{gathered}[/tex]So the required solution is,
[tex]y=\frac{x-2}{13}[/tex]what is the profit in dollar and cents that the store makes per sweater?p=26s
Given the expression:
[tex]p=26s[/tex]if 'p' represents the profit in dollar and 's' represents the number fo sweaters that the store sells, then we have that the store makes 26 dollars per sweater.
We can see this since the equation is the same as a linear function with rate of change m:
[tex]y=mx[/tex]An observer stands 400 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 300 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 300 ft from the ground?
Let's draw the scenario to understand it better:
From the figure, the information given are:
[tex]\frac{dy}{d\text{t}}=\text{ 300 ft./sec}[/tex]Question:
[tex]\text{What is }\frac{dD}{dt}\text{ at y = }300\text{ ft.}[/tex]Step 1: We write a function that relates the quantities in the diagram using Pythagorean
Theorem.
[tex]\text{ 400}^2\text{ + }y^2=D^2[/tex]Step 2: Differentiate with respect to t.
[tex]2y\frac{dy}{dt}=\text{ }2D\frac{dD}{dt}[/tex]Now we wish to plug in specific numbers for every quantity in the above equation except for dD/dt. However, we notice that we don’t have a specific value for D at y = 400 ft. So first we need to find D at y = 400 ft. using the Pythagorean Theorem.
[tex]400^2\text{ + }300^2=D^2[/tex][tex]D\text{ = }\sqrt[]{400^2+300^2}[/tex][tex]D\text{ = 500 ft.}[/tex]Step 3: Finish the problem by plugging in numbers for every quantity in the equation
containing dD/dt.
[tex]2(300)(300)\text{ = 2(500)}(\frac{dD}{dt})[/tex][tex]\frac{dD}{dt}=\text{ }\frac{2(300)(300)}{2(500)}[/tex][tex]\frac{dD}{dt}=\text{ }\frac{180,000}{1000}[/tex][tex]\frac{dD}{dt}=180[/tex]Conclusion: When the rocket is 300 ft. feet from the ground, the distance between the observer and the rocket is increasing at a rate of 180 ft./sec.
Points XX and ZZ are on a number line, and point YY partitions \overline{XZ}
XZ
into two parts so that the ratio of the length of \overline{XY}
XY
to the length of \overline{YZ}
YZ
is 5:75:7.
The coordinate of XX is 1.31.3, and the coordinate of YY is 3.83.8. What is the coordinate of ZZ?
If a point divides two other points in any ratio m : n, then the coordinate of the point dividing the given two points can be known using section formula. The coordinate of the point ZZ on the number line is (4.06, 8.46).
What is section formula?Section formula is used in Coordinate geometry to find the ratio between two points on a line given one another point on the line.
Given that,
The ratio XZ : XY : YZ = 5 : 75 : 7
The coordinates of XX are (1.31, 3) and of YY are (3.83, 8).
Since YY is partitions the point XX and ZZ. In order to find the the coordinates of ZZ section formula can be used as below,
x = (m₁x₂ + m₂x₁) ÷ (m₁ + m₂) and y = (m₁y₂ + m₂y₁) ÷ (m₁ + m₂)
Here, m₁ : m₂ is equal to XY : YZ which can be known from the given ratio as,
XY : YZ = 75 : 7
=> m₁ : m₂ = 75 : 7
=> m₁ = 75 and m₂ = 7.
Suppose the coordinates of XX, YY and ZZ are (x₁, y₁), (x, y) and (x₂, y₂) respectively.
Substitute the respective values in the section formula to get,
3.83 = (75x₂ + 7 × 1.31) ÷ (75 + 7)
=> (75x₂ + 7 × 1.31) = 82 × 3.83
=> 75x₂ = 82 × 3.83 - 7 × 1.31
=> x₂ = (82 × 3.83 - 7 × 1.31) ÷ 75
=> x₂ = 4.06
And,
8 = (75y₂ + 7 × 3) ÷ (75 + 7)
=> (75y₂ + 7 × 3) = 82 × 8
=> 75y₂ = 82 × 8 - 7 × 3
=> y₂ = (82 × 8 - 7 × 3) ÷ 75
=> y₂ = 8.46
Thus, ( x₂, y₂) is equal to (4.06, 8.46).
Hence, the coordinate of ZZ is given by (4.06, 8.46).
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Which of the following is the value of y that solves the system of equations shown below?
(1) -5
(3) 6
3x + 4y = 16
(2) 8
(4) -2
x + y =16
Answer:
(4) -2
Explanation:
Given the system of equations:
[tex]\begin{gathered} 3x+4y=16 \\ x+y=6 \end{gathered}[/tex]Set x as the subject in the second equation: x=6-y
Substitute x=6-y into the first equation:
[tex]\begin{gathered} 3x+4y=16 \\ 3(6-y)+4y=16 \\ 18-3y+4y=16 \\ y=16-18 \\ y=-2 \end{gathered}[/tex]The value of y that solves the system of equations is -2.
identify the transformation from the pre-made to the image
You can notice that each point of both images are at the same distance of the origin of coordinates. It means that the transfromation is a reflection across the line y = x
answer: reflection across y = x
what is the flying distance between the greenhouse and the stadium
The flying distance between the greenhouse and the stadium = 5 units
Explanations:The coordinates of the greenhouse: (-6, 0)
The coordinates of the stadium: (-2, 3)
The distance between two points of coordinates (x₁, y₁) and (x₂, y₂) is given as:
[tex]D\text{ = }\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]For the flying distance between the greenhouse and the stadium:
x₁ = -6, y₁ = 0, x₂ = -2, y₂ = 3
Substitute these values into the distance equation given above:
[tex]\begin{gathered} D\text{ = }\sqrt[]{(-2-(-6))^2+(3-0)^2} \\ D\text{ = }\sqrt[]{(-2+6)^2+3^2} \\ D\text{ = }\sqrt[]{4^2+3^2} \\ D\text{ = }\sqrt[]{16+9} \\ D\text{ = }\sqrt[]{25} \\ D\text{ = 5} \end{gathered}[/tex]The flying distance between the greenhouse and the stadium = 5 units
I'll send pic of equation
Given the following functions:
f(x) = 3x
g(x) = x + 4
h(x) = x^2 - 1
Before simplifying g(f(-1)), let's first determine f(-1).
f(x) = 3x
f(-1) = 3(-1)
f(-1) = -3
g(x) = x + 4
g(f(-1)) = (-3) + 4
= -3 + 4
g(f(-1)) = 1
Therefore, the answer is 1.
Find the slope of the line.
The slope of the line is [tex]-\frac{1}{3}[/tex].
Consider any two points on the graph,
Let A(x1,y1) and B(x2,y2) are the two points on graph as shown in figure.
The two points are A(-2,0) and B(-4,6)
Slope of a line can be calculated using formula,
[tex]Slope=\frac{y2-y1}{x2-x1}\\\\Slope=\frac{-4-(-2)}{6-0}\\ \\Slope=\frac{-4+2}{6}\\ \\Slope=\frac{-2}{6}\\ \\Slope=\frac{-1}{3}[/tex]
Thus, the slope of the line is [tex]-\frac{1}{3}[/tex].
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Are The Ratios 1:2 and 5:14 equivalent?
The ratios are equivalent if:
[tex]\frac{2}{1}\equiv\frac{14}{7}[/tex]For example, given a length of 20cm:
[tex]\begin{gathered} 20\times\frac{2}{1}=20\times2=40\operatorname{cm} \\ 20\times\frac{14}{7}=20\times2=40\operatorname{cm} \end{gathered}[/tex]Since:
[tex]40\operatorname{cm}=40\operatorname{cm}[/tex]We can conclude that they are equivalent
The one-to-one functions g and h are defined as follows.g(x) = 4x - 3h={(-6, 3), (-4, 7), (3, -8), (6, 4)Find the following
It is given that
[tex]g(x)=4x-3[/tex]Now to find the inverse of g
Let
[tex]\begin{gathered} y=4x-3 \\ 4x=y+3 \\ x=\frac{y+3}{4} \end{gathered}[/tex]So
[tex]g^{-1}(x)=\frac{x+3}{4}[/tex]Now we know that
[tex](g^{-1}.g)(x)=\text{ x}[/tex]So
[tex](g^{-1}.g)(2)=2[/tex]And since
[tex]h(-6)=3[/tex]So
[tex]h^{-1}(3)=-6[/tex]Write the expression in simplest form.[tex] \sqrt{49 {x}^{5} } [/tex]
Answer:
[tex]=7x^2\sqrt[]{x}[/tex]Explanation:
Given the expression:
[tex]\sqrt{49x^5}[/tex]First, the expression can be rewritten in the form below:
[tex]\begin{gathered} =\sqrt[]{49\times x^5} \\ By\text{ multiplication of surds: }=\sqrt[]{mn}=\sqrt[]{m}\times\sqrt[]{n} \\ =\sqrt[]{49^{}}\times\sqrt[]{x^5} \end{gathered}[/tex]Next, we simplify:
[tex]\begin{gathered} =\sqrt[]{7^2}\times\sqrt[]{x^4\times x} \\ =7\times\sqrt[]{x^4}\times\sqrt[]{x} \\ =7\times x^2\times\sqrt[]{x} \end{gathered}[/tex]The simplest form of the expression is:
[tex]=7x^2\sqrt[]{x}[/tex]Graph the following function using the techniques of shifting, compressing, stretching and or reflecting.
Answer:
Domain: (-∞, ∞)
Range: (-∞, ∞)
Explanation:
The parenting function of g(x) = (x + 3)³ + 5 is f(x) = x³.
The graph of f(x) = x³ is
Then, if we have a function g(x) = f(x + c), we can say that g(x) is the graph of f(x) shifted c units to the left and if we have a function g(x) = f(x) + c, we can say that g(x) is the graph of f(x) shifted c units up
In this case, g(x) = f(x + 3) + 5 because
g(x) = f(x + 3) + 5
g(x) = (x + 3)³ + 5
So, g(x) has the graph of f(x) shifted 3 units to the left and 5 units up.
Therefore, the graph of g(x) is
Now, the domain is the set of values that x can take and the range is the set of values that y can take, so the domain and range are all the real number.
Domain: (-∞, ∞)
Range: (-∞, ∞)
Suppose that the dollar value v(t) of a certain car that is t years old is given by the following exponential function. v(t) = 21, 300 * (1.24) ^ t
Given:-
[tex]v(t)=21300(1.24)^t[/tex]To find the initial value, does the fucnction represent growth or decay.
Here the value of a is 21300.
So we get,
[tex]1+r=1.24[/tex]So the value of r is,
[tex]r=0.24[/tex]So now we get percentage is,
[tex]24\%[/tex]Rate of change is 24%.
So the required solution is,
[tex]initial\text{ value =21300}[/tex]So the percentage range is,
[tex]24\%[/tex]
QuesQuestion 3 (1 point)Let sin (47) = 0.7314. Enter an angle (B), in degrees, where cos (B) = 0.7314.Answer:Blank 1:Next PageBack
Solution
Let sin (47) = 0.7314. Enter an angle (B), in degrees,
[tex]sin(47)=0.7314[/tex]where cos (B) = 0.7314 means
[tex]\begin{gathered} sin(A)=cos(90-A) \\ let\text{ A = 47} \\ sin(A)=cos(90-47) \\ sin(A)=cos43 \end{gathered}[/tex]since the vaule of sin(A) = 0.7314
Hence the cos(B) = cos43 = 0.7314
The angle of B = 43°
Given any triangle ABC labeled as shown, the law of sines states:BaсССAСAbbsin Asin BsincΟ Α.сasin Asin BB.sincsincbaсsincsin BO c.sin Abaсsin Asin BsincD.aС
Simply, the law of sine states that the ratio of the lenght of a side of a triangle to the sine of the angle opposite that side is the same for all side and angles in a given triangle. In other words,
[tex]\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}[/tex]which corresponds to option D
i inserted a picture of the question which is question 7, i can give you the answer to the previous question which is question 6 if it helps
The first thing we have to know is that the trinomial (polynomial of three terms) is called a perfect square trinomial, the polynomial that when factorized gives us a perfect squared expression in the following way
[tex]a^2+2ab+b^2=(a+b)^2[/tex][tex]\begin{gathered} w^2-3w=350 \\ a=1 \\ 2ab=-3 \\ b=-\frac{3}{2} \\ b^2=\frac{9}{4} \end{gathered}[/tex][tex]\begin{gathered} w^2-3w-\frac{9}{4}=350-\frac{9}{4} \\ w^2-3w-\frac{9}{4}=347.75 \\ (w-\frac{3}{2})^2=347.75 \end{gathered}[/tex][tex](w-\frac{3}{2})^2-\frac{1409}{4}=0\to\text{answer}[/tex]Which transformation produces a similar but incongruent figure?
Similar means that the figure has the same shape, but not neccesarily the same size. It is said, also, that it is incoungruent, so it definetly doesn't have the same size. A transformation that gives you a similar but incongruent figure is a dialation, rotation and/or translation. I'll do a drawing to illustrate:
Notice that I've done the three of them. I have the same figure, smaller, in another position and rotated.
If LM = 14 and MN = 11, what is KM?Write your answer as a whole number or as a decimal rounded to the nearest hundredth
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
LM = 14
MN = 11
KM = ?
Step 02:
We must apply the triangle rules to find the solution.
Pythagoras Theorem
LM² = LN² + MN²
(14)² = LN² + (11)²
(14)² - (11)² = LN²
[tex]LN\text{ = }\sqrt[]{196-121}[/tex]LN = 8.66
[tex]\frac{LN}{MN}=\frac{KN}{LN}[/tex][tex]\frac{8.66}{11}=\frac{KN}{8.66}[/tex]8.66 * 8.66 = 11 * KN
6.818 = KN
KM = KN + MN
KM = 6.818 + 11 = 17.818
The answer is:
The value of KM is 17.82
Donovan has 5 times as many as red fish as he does blue fish and half as many gray fish as blue fish. How many total fish (F) does Donovan have?answer optionsF = 6x + 1/2xF = 1/2x + 5 + xF = 6 + 1/2xF = 5x + 1/2x
Donavan has 6x + 1/2 x total number of fishes.
Let Donovan has x number of blue fishes.
Hence the number of red fishes = 5x
The number of gray fishes = 1/2 of x = 1/2 x
Total number of fishes = red + blue + gray = 5x + x + 1/2 x = 6 + 1/2 x
For something like an equation to have any importance, the coefficient involving at least any variable must not be 0. In actuality, the following equation would either be inconsistent (for b ≠ 0) and have no solution, or full n-tuples are solutions, as was mentioned for one variable, if any variable does actually have a zero coefficient.
A regular polynomial over such a field, from which the coefficients are drawn, can also be equalized to zero to produce a linear equation. Since linear equations typically do a good job of representing non-linear systems, they are used extensively in all branches of mathematics as well as in physics and engineering.
Therefore we can infer that Donavan has 6x + 1/2 x total number of fishes.
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Please help me with this problem for my son to understand thank you.Factor completely.36x^4+18x^3+40x^2Enter your answer in the box.
The expression is given to be:
[tex]36x^4+18x^3+40x^2[/tex]Rewrite the terms so that we can factorize the common terms out:
[tex]\begin{gathered} 36x^4\Rightarrow2x^2\cdot18x^2 \\ 18x^3\Rightarrow2x^2\cdot9x \\ 40x^2\Rightarrow2x^2\cdot20 \end{gathered}[/tex]Therefore, the expression can be rewritten to be:
[tex]2x^2\cdot18x^2+2x^2\cdot9x+2x^2\cdot20[/tex]Since there is a common term, 2x², we can factorize to give:
[tex]\Rightarrow2x^2(18x^2+9x+20)[/tex]The quadratic function in the parentheses cannot be factored further.
Therefore, the answer is:
[tex]2x^2(18x^2+9x+20)[/tex]2x-8=4Wouldn't the answer be 10
We need to sum 8 to both sides of the equation:
[tex]undefined[/tex]how do I find the linear change
In order to find the rate of change of the given line, use the following formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]where (x1,y1) and (x2,y2) are two points of the line.
From the given graph you can select any pair of points, for example:
(x1,y1) = (2,400)
(x2,y2) = (4,1200)
replace the values of the previous coordinates in the formula for m:
[tex]m=\frac{1200-400_{}}{4-2}=\frac{800}{2}=400[/tex]Hence, the rate of change is 400
Write the number as the product of a real number and i[tex] \sqrt{ - 24} [/tex]