- Roll a die twice and consider the events A = {first roll gives at
least 4}, B = {second roll gives at most 4} and C = {the sum of the
rolls is 10}.
(a) Find P(A), P(B),P(C), and P(A ∩ B ∩ C).
(b) Are A, B, and C independent?

Using the formula of combination and the provided condition that any child can get any number of books, The answer is 637.

Combinations are mathematical operations that count the number of potential configurations for a set of elements when the order of the selection is irrelevant. We use permutation to determine how many possible ways we have to arrange items from a collection system.

The required formula is:

[tex]C(n,r) = \frac{n!}{r!(n-r)!}[/tex]

where, n= total number of objects in the set.

r= number of choosing objects from the set.

from the given data, total number of books = n= 5

number of children we need to distribute, r= 10

total possible ways of distribution= C(10,1)+C(10,2)+C(10,3)+C(10,4)+C(10,5)

=10+45+120+210+252

=637

hence, we have total 637 possible ways to distribute.

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The greatest common factor between the given two numbers will be 20.

The greatest common factor is the largest positive number which can be divided into given numbers evenly.

Given that the two numbers are 1240 and 6660. First, factorize each number and find the common multiple of both numbers.

1240 = 2 x 2 x 2 x 5 x 31

6660 = 2 x 2 x 3 x 3 x 5 x 37

The common numbers are,

2 x 2 x 5

4 x 5

20

Therefore, the greatest common factor between the given two numbers will be 20.

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The total years taken to double the money according to Compounding will be 13.75 years.

What is compound interest?

Compound interest is that the interest on savings calculated on each the initial principal and therefore the accumulated interest from previous periods.

Main body:

Given :

rate = 2%

amount = $1700

Total amount = $2200

Compounded monthly,

so formula becomes,

Total amount = amount (1+r/400)^4t

T = A([tex](1+r/400)^{4t[/tex]

2200 = 1700(1 +2/400)^4t

22/17 = (1 +1/200)^4t

22/17 = (1.005)^4t

1.29 = (1.005)^4t

taking log on both sides

0.11/0.002 = 4t

t = 13.75 years.

Hence the time taken to double the money is 13.75 years.

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Answer:

3= 26

4= 11 because it is area of triangle

Answer:

6

Step-by-step explanation:

Answer:

Angle-Side-Angle or also known as ASA

Step-by-step explanation:

To find out how far the seagull flew while the sailor was watching, you can use the following steps:

Calculate the total distance the seagull flew in 28 seconds. You can do this by multiplying the seagull's speed (11 meters per second) by the time it flew (28 seconds):

Distance = Speed * Time

= 11 meters/second * 28 seconds

= 308 meters

Calculate the time the sailor watched the seagull fly. You can do this by dividing the total time the seagull flew (28 seconds) by 4:

Time = 28 seconds / 4

= 7 seconds

Calculate the distance the seagull flew while the sailor was watching. You can do this by multiplying the seagull's speed (11 meters per second) by the time the sailor watched the seagull fly (7 seconds):

Distance = Speed * Time

= 11 meters/second * 7 seconds

= 77 meters

Therefore, the seagull flew 77 meters while the sailor was watching.

What are conditional problems?

Conditional problems are problems that involve determining the likelihood or probability of an event occurring under certain conditions. They often involve the use of probability and statistics to evaluate the likelihood of different outcomes or to make predictions about future events.

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[tex]~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{"a"~is~negative}{op ens~\cap}\qquad \stackrel{"a"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x=3\\ y=-3 \end{cases}\implies y=a(x-3)^2 - 3\hspace{5em}\textit{we also know that} \begin{cases} x=2\\ y=-1 \end{cases} \\\\\\ -1=a(2-3)^2 -3\implies 2=a(-1)^2\implies 2=a~\hfill \boxed{y=2(x-3)^2 -3}[/tex]

y-intercept of given function is y = 1

How to graph the function?

The collection of ordered pairings with the (x,y). (x,y) sign is known as the graph of a function in mathematics, where  f(x)=y. x and  f(x) are Cartesian coordinates of points in two-dimensional space, and in the typical scenario where they are real numbers, they constitute a subset of this plane.

Given function , f(x) = 2· [tex]3^{x}[/tex]

Graph of the function is given below:

As the given graph is intersecting the y-axis at y=1 So, y-intercept of given function is y = 1

Therefore, y-intercept is y = 1

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Answer:

Step-by-step explanation:

Let's call the speed of the boat in the river x miles per hour. The boat travels 2 miles downstream in the same amount of time it takes to travel 2 miles upstream, so the speed of the boat relative to the water is the same in both directions. We can set up the following equation to represent this relationship:

21 - x = x + current speed

Solving for x, we find that the speed of the boat in the river is (21 - x)/2 = 10.5 miles per hour.

We can then use this value to find the current speed of the river. The boat travels 2.6 miles downstream in the same amount of time it takes to travel 2 miles upstream, so the speed of the boat relative to the ground is the same in both directions. We can set up the following equation to represent this relationship:

10.5 + current speed = 10.5 - current speed

Solving for the current speed, we find that it is (10.5 - 10.5)/2 = 0.

Therefore, the current speed of the river is 0 miles per hour.

As ΔADE and ΔABC are similar because of their slope so correct option is A.

What are similar triangles?

Similar triangles square measure triangles that have identical form, however their sizes might vary. All equal triangles, squares of any facet lengths square measure samples of similar objects.

Main Body;

let co-ordinates of the points

A = (x₁ ,y₁ ) E =(x₂, y₂)

D =( x₃,y₃) C= (x₄ ,y₄)  B =( x₅,y₅)

Now, Slope of a line = Δy/Δx

.: Slope of line AD = y₃ -y₂/x₂-x1 .

                              = DE/AE

... Slope of line AB = y₅-y₄/x₄-x₁ = BC/ AC

Since ΔADE ~ ΔABC

Now , Slope of AD = slope of AB

=> DE/ AE = BC /AC

In option A

By property of Similar triangle -

DE /BC  = BC /АC

Hence, option A is correct answer.

All other options are incorrect

Since Sides of a similar triangle may not be equal.

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A table of values that models this situation is as follows;

Distance (miles)               Time (hour)

120                                          0

150                                          1

180                                          2

210                                          3

240                                          4

An equation for D in terms of t representing Elizabeth's distance from city A, t hours after leaving her house is d = 30t + 120.

Mathematically, a constant of proportionality (time) for Elizabeth's journey (distance) with respect to time can be represented with the following mathematical expression:

d = kt

Where:

Therefore, the required function that relates the distance (d) and speed (s) at t hours is given by;

d = 30t + 120

Next, we would develop a table of value;

At time (t) = 1 hour, we have;

Distance, d = 30(1) + 120

Distance, d = 150 miles.

At time (t) = 2 hours, we have;

Distance, d = 30(2) + 120

Distance, d = 180 miles.

At time (t) = 3 hours, we have;

Distance, d = 30(3) + 120

Distance, d = 210 miles.

At time (t) = 4 hours, we have;

Distance, d = 30(4) + 120

Distance, d = 240 miles.

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It is Syllogism. It is an example of a type of reasoning where a conclusion is reached from two given or assumed propositions, each of which shares a term with the conclusion.

The Greek term "syllogismos," which means "conclusion, inference," is the source of the English word "syllogism." A logical argument employing assertions and deductive reasoning is known as a syllogism. The most important contribution to the topic of syllogisms is attributed to Aristotle. A major premise, a minor premise, and a conclusion make up a syllogism. One phrase from each of the premises and the conclusion are shared.

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Relationship to the line y=3/4x and its equation is line B.

What is an equation ?

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

y = mx+c

y = 3/4x [which shows that the line B as the points]

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Answer:

Step-by-step explanation:

2,400 consoles

The time it takes the trains 2.2 hours to be 396 miles apart.

Time is the duration of an event.

Since two trains leave the station at the same time, one heading west and the other east. The westbound train travels at 85 miles per hour. The eastbound train travels at 95 miles per hour.

Let

Also,

Since both trains move in opposite directions and leave at the same time, their distance apart after time t is D' = d + D

= vt + Vt

= 85t + 95t

= 180t

Since D' = 396 miles, we have that

396 = 180t

t = 396 m/180 mph

t = 2.2 hours

So, it takes the trains 2.2 hours to be 396 miles apart.

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The area of the rectangle training field is 4264 m².

We know the perimeter of any 2D figure is the sum of the lengths of all the sides except the circle and the area of a rectangle is the product of its length and width.

We know the area of a rectangle is (length×width).

So, The area of the rectangle 82m long and 52m wide is,

= (82×52) m².

= 4264 m².

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Step-by-step explanation:

32^2-24^2=448

square root 448=21

the small and large parallel line is 21=21

24+9=33

33^2+21^2=1530

square root 1530=39

39-32=7

~7~=6.75

x=6.75

The line parallel to line y= -x + 7  is y= -x + 9.

What is slope intercept form?

The intercept formula y = mx + b is used when you know the slope of the line you are examining, and the specified point is also the y-intercept (0, b). In this formula, b represents the y-value of the y-intercept.

Solution: We will use slope intercept formula y= mx + b

Given that the parallel line passes through (4,5), which means x=4 and y=5

Also, m = -1

Putting value of x and y in slope intercept form to find b, we get

y=mx + b

⇒ 5= (-1x4) + b

⇒ 5= -4 +b

⇒5 + 4= b

⇒b=9

Now putting value of b and m in slope intercept form, we get the parallel line

y= -x + 9

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The elements of the required sets would be

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

What are sets?

A set is a mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

Given:

Let the universal set is (S) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

P = {Odd numbers}  = {1, 3, 5, 7, 9}

Q = {Prime Numbers} = {2, 3, 5, 7}

R = {1, 2, 3, 4, 5}

So by using the properties of sets we can find

P' = {2, 4, 6, 8, 10}

Q' = {1, 4, 6, 8, 9, 10}

R' = {6, 7, 8, 9, 10}

P∩Q = {3, 5, 7}

R'∪P' = {2, 4, 6, 7, 8, 9, 10}

P∪R = {1, 2, 3, 4, 5, 7, 9}

P∩R = {1, 3, 5}

P∪Q∪R = {1, 2, 3, 4, 5, 7, 9}

P'∩Q'∩R' = S - P∪Q∪R

                = {6, 8, 10}

Now we can find the elements of the required sets:

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

Hence, the elements of the required sets would be

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

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Answer:

[tex]\huge{\boxed{46 \text{ or } 67 \text{ or } 88 \text{ degrees}}}[/tex]

Step-by-step explanation:

An isosceles triangle is a triangle in which two of the sides and angles are equal.  Because the sum of all of the angles in a triangle is 180°, you may setup the following equation.

46 + x + z = 180°

Depending on the following scenario, your answer may be one of two options.  If the "third angle" refers to the angle that is not at the base of the triangle, then your equation is as follows:

z = x

46 + x + x = 180

46 + 2x = 180

2x = 134

x = 67

If the third angle is one of the base angles and x is a base angle, then x is also 46, because base angles are equal.

If the third angle is one of the base angles and x is the third angle, then your equation is as follows:

z = 46

46 + 46 + x = 180

92 + x = 180

x = 88

Hope it helps :) and let me know if you want me to elaborate.

Answer:

1.3 is the common difference.

Step-by-step explanation:

The number of ways the 6 books can be arranged on the 4 shelves by Amanda is 360 ways

From the question, we have the following parameters that can be used in our computation:

Total number of books, n = 6

Total number of shelves, r = 4

The number of ways the books could be arranged on the shelves is calculated using the following permutation formula

Total = ⁿPᵣ

Where

n = 6 and r = 4

Substitute the known values in the above equation, so, we have the following representation

Total = ⁶P₄

Apply the permutation formula

ⁿPᵣ = n!/(n - r)!

So, we have the following representation

Total = 6!/2!

Evaluate the quotient

Total = 360

Hence, the number of ways is 360

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The perimeter of the rectangle can be expressed as 12x+2

The perimeter of a rectangle is the total distance covered by its boundaries or the sides. Since there are four sides of a rectangle, thus, the perimeter of the rectangle will be the sum of all four sides.

The perimeter of a rectangle is given as 2(l+w)

where l is the length and w is the width.

If the length is represented by 4x and width by 2x+1, therefore the perimeter of the rectangle = 2(4x+2x+1)

= 2( 6x+1)

= 12x+2

the perimeter can also be found by adding all the four sides, this means that

perimeter = 4x+4x+2x+1+2x+1=8x+4x+2= 12x+2

therefore the perimeter of the rectangle is 12x+2.

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Olivia's quiz average must be 24.4 to achieve a 90 average in algebra.

To get the average for the quiz, we first evaluate for the sum of the test average and the homework average, and the subtract the result from 90 average in algebra.

the test average = (92/100) × 55

test average = 50.6

100% homework average implies Olivia got a homework average of 15

Let x be the quiz average, so that

x = 90 - (50.6 + 15)

x = 90 - 65.6

x = 24.4

Therefore, the quiz average of 24.4 is what Olivia needs to achieve a 99 average in algebra.

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After solving, the value of f^(-1)(g(7)) is 6.

In the given question, we have to find the f^(-1)(g(7)).

The given table of functions f and g is:

x          f(x)                   y          g(x)  

-3          1                     -2          -5

0          4                     1            -4

1           5                     5            2

6         -6                     7           -6

To find the value of f^(-1)(g(7)) we put the value of g(7) from the table. As we can see in the table that at the value of g(x) at point 7 is -6.

Then we find the value of from the table of f(x).

f^(-1)(g(7)) = f^(-1)(-6)

The value of f^(-1)(-6) from the table f(x) is

f^(-1)(g(7)) = 6

Hence, the value of f^(-1)(g(7)) is 6.

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Answer:

10-x

Step-by-step explanation:

Answer: 11

Step-by-step explanation:

we know that : ax² + bx + c = 0

then,

a = −1

b = −22

c = −121

putting the values in the following equation:

x=−b±√b²−4ac ⁄ 2a

=−(−22)±√(−22)²−4⌈(−1)(−121)⌉  ⁄  2(−1)

=22±√484−484  ⁄ −2

=22±√0  ⁄ −2

=22 ⁄ −2

=11 ⁄ −1

=11

Hence the answer for this equation will be 11.

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