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Using Pascal's triangle we can find the binomial coefficient, by which we get 1m^4 + 16m^3 + 96m^2 + 128m + 256

Pascal's triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. It is named after the French mathematician Blaise Pascal, who studied its properties in the 17th century. The rows of the triangle are numbered starting from 0, and the first number in row n is always 1. The other numbers in row n are the sum of the corresponding number in row n-1 and the number directly above it. The nth number in the kth row of Pascal's triangle is denoted by C(n,k) and it is called binomial coefficient, it is also denoted by nCk. It is used in many areas of mathematics, particularly in combinatorics and algebra, and it can be used to expand binomial expressions using the binomial theorem.

The binomial theorem states that for any non-negative integer n and any binomial expression (a+b)^n, the expansion is given by:

(a+b)^n = C(n,0)a^n*b^0 + C(n,1)a^(n-1)*b^1 + C(n,2)a^(n-2)b^2 + ... + C(n,n)a^0b^n

where C(n,k) = n!/(k!(n-k)!) is the binomial coefficient, which can be found using Pascal's triangle.

Using this theorem:

(m+4)^4 = C(4,0)m^44^0 + C(4,1)m^34^1 + C(4,2)m^24^2 + C(4,3)m^14^3 + C(4,4)m^0*4^4

Using Pascal's triangle we can find the binomial coefficient:

C(4,0)=1, C(4,1)=4, C(4,2)=6, C(4,3)=4, C(4,4)=1

So the expanded form of (m+4)^4 is:

1m^4 + 4m^34 + 6m^24^2 + 4m4^3 + 14^4

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Answer:

Step-by-step explanation: add i'll the sides togetheir

On adding three digit number 109 to 7845 we get 7954 so the number is 109.

The one -ended or all of the addends in a 3-digit addition are numbers of three digits. Three-digit numbers can be added by placing them in columns so according their place values. Following that, we start adding the numbers from of the ones and tens columns before finishing with the hundreds column.

The numbers being added are termed addends, and the outcome of the operation, or the answer we ultimately get, is referred to as the sum. It is a fundamental mathematical operation that we employ on a daily basis. We add numbers in a variety of circumstances.

7845 + 1100

= 8945 so it is wrong

7845+109=7954

109 is 3 digit

and because of that, we won't need to consider any other options. If you checked them again, you would not receive a three-digit number.

so 109 is your answer

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Note that the explicit equation and domain for an arithmetic sequence with a first term of 3 and a second term of 5 is n = 3 + 2(n − 1); all integers where n ≥ 1 (Option D)

Arithmetic progression or arithmetic sequence is a numerical series in which the difference between subsequent terms is constant. For example, the sequence 5, 7, 9, 11, 13, 15..

The first term of the arithmetic sequence is 3, followed by 5. Thus, the common difference is, 5-3  = 2. You use these values to the derived formulas for arithmetic progression. One of these formulas is

An = A1 + (n-1)d

where

An is the nth term in the sequence

A1 is the first term of the sequence

n is the total number of terms within the sequence

d is a common difference.

In this problem, A1 = 5 and d=2. Substituting to the equation,

An = 3 + (2)(n-1)

An = 3 + 2(n-1), where n should be more than 1. In order for it to become a sequence, it should contain more than one term.

Therefore, it is correct to appropriate an explicit equation and domain for an arithmetic sequence with a first term of 3 and a second term of 5 to be: n = 3 + 2(n − 1); all integers where n ≥ 1

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Answer: y = -3x - 6

Step-by-step explanation: y - 9 = ((9 + 3)/(-5 + 1))(x + 5)
y = 9 - 3(x + 5)

y = 9 - 3x - 15

y = -3x - 6

It is not possible to construct a triangle whose sides are 5 cm 7 cm and 12 cm because the sum of two sides(5 cm and 7 cm) are equal to the third sides(12 cm). Two equal sides cannot form a triangle.

In the given question, we have to check that it is possible to construct a triangle whose sides are 5 cm 7 cm and 12 cm.

The sum of the two sides must always be greater than the third side in order to determine whether or not the given sides may form a triangle.

To check this we firstly add the 5 and 7.

7 + 5 = 12

12 = 12

Now we add 5 and 12

5 + 12 > 7

17 > 7

Now we add 7 and 12

7 + 12 > 5

19 > 5

It is not possible to construct a triangle whose sides are 5 cm 7 cm and 12 cm because the sum of two sides(5 cm and 7 cm) are equal to the third sides(12 cm). Two equal sides cannot form a triangle.

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The complete question is:

Can a triangle have the sides 5 cm 7 cm and 12 cm?

The value of b can be -32 or +32 to ensure that the equation has only one true solution.

The discriminant is the quadratic formula section behind the square root symbol: b2-4ac. The discriminant indicates whether there are two solutions, one solution, or none at all. The "discriminant" is the argument (that is, the contents) of the square root, which is the equation b2 4ac, because by utilizing its value, you may "discriminate" between (that is, determine the difference between) the various solution types.

Here,

The discriminant of the equation is ...

d = b² -4ac

d = b² -4(2)(128)

d = b² -1024

In order for there to be one real solution, we must have d=0. So ...

0 = b² -1024

1024 = b²

±√1024 = ±32 = b

The value of b can be -32 or +32 to cause the equation to have one real solution.

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The magnitude of the correlation coefficient indicates the strength of the association. For example, a correlation of r = 0.9 suggests a strong, positive association between two variables, whereas a correlation of r = -0.2 suggest a weak, negative association.

How to interpret a correlation coefficient?

The statistical measure of how well changes in the value of one variable predict changes in the value of another is called a correlation coefficient. The value rises or falls together when the variables are positively linked.

Is r or r 2 the correlation coefficient?

The summary table in the Regression output contains the "R" value, which is the coefficient of correlation. The term "coefficient of determination" also applies to R square. To obtain the value of R squared, multiply R by R. In other words, the square of the coefficient of correlation is the coefficient of determination.

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Substitution is the name of the process in algebra which is used for swapping an algebraic letter for its value.

The substitution method is the algebraic method to solve different linear equations. In this method, the value of one variable from one of the equations is substituted in the other equation.

In substitution the following steps are used to solve an equation by substitution method:

Step 1. Simplify the given equation by removing the parenthesis.

Step 2. Solve one of the equations for any unknown value.

Step 3. Substitute step 2 result in the other equation.

Step 4. Now solve the equation which gets from step 3 obtained using elementary arithmetic operations.

Step 5. Finally, solve the final equation for the second variable.

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Answer:

The principle of factorization is that every square matrix can be expressed as a product of a lower triangular matrix and upper triangular matrix. On the basis of this fact, these lower and upper triangular matrices help us in finding the unknowns.

Step-by-step explanation:

Hope it helps!

0 is not an exponential function.

We know that the general form of exponential function f with base b is defined as:

f(x) = b^x

where b > 0 and b is not equal to 1

and x is any real number.

If it is negative  b < 0, the function is undefined for -1 < x < 1.

If in an exponential function the base is less than zero, then it would make the function values complex for certain fractional values.

If the base of exponential function is 0 then function would create a graph which violates the properties / definition of exponential functions.

Also, we can not write 0 in exponential function form.

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Answer:The 3 in 103 is called the exponent. The expression 103 is called the exponential expression. base → 103 ←exponent

Step-by-step explanation:

The quote mentioned in the question was given by John F. Kerry who was a famous American personality, he was born on 11 December 1943.

Professionally he was an attorney and later became a politician and, diplomat and he currently serves as the first United States special presidential envoy for climate. He is a member of the Forbes family and the Democratic Party, and that he previously served as the 68th United States secretary of state .

Some of his other popular quotes are , "I saw courage both in the Vietnam War and in the struggle to stop it. I learned that patriotism includes protest, not just military service."

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The link between s, t, and u is therefore u2= t2+s2 and provides the answer to the triangle problem.

A triangle is a polygon since it has three parts and three vertices. It is a basic geometric shape. A triangle with the corners A, B, & C is referred to as a triangle ABC. A singular plane & triangle in Geometric shapes are found so when three components are not collinear. Due to its right side and three corners, triangles are considered polygons. The corners of a triangle are defined as the locations at which its three sides meet. 180 degrees is obtained by multiplying three triangle angles.

Here,

Given: Triangle STU is a right triangle in this instance.

Thus,

We learn:

parts s, t, and u

Their connection is as follows:

Given that it's a right triangle

So,

The Pythagoras Theorem states that

=>u²= t²+s²

The link between s, t, and u is therefore u2= t2+s2 and provides the answer to the triangle problem.

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The derivatives of the given functions are 2 - 7/x, 6x + 4 - 5/x^4 Inx, -10t + 3 - 1/t Log t, and 6/x In 5 - 4x, respectively.

5. f'(x) = 2 - 7/x;

6. f'(x) = 6x + 4 - 5/x^4 Inx;

7. y' = -10t + 3 - 1/t Log t;

8. y' = 6/x In 5 - 4x

5. To find the derivative of the given expression, apply the chain rule. The derivative of 2x is 2, and the derivative of -7Ln x is -7/x. Thus, the derivative of the function is 2 - 7/x.

6. To find the derivative of the given expression, apply the chain rule. The derivative of 3x^2 is 6x, the derivative of 4x is 4, and the derivative of -Inx^5 is -5/x^4 Inx. Thus, the derivative of the function is 6x + 4 - 5/x^4 Inx.

7. To find the derivative of the given expression, apply the chain rule. The derivative of -5t^2 is -10t, the derivative of 4 is 0, and the derivative of -3t is -3. The derivative of Logt is 1/t Log t. Thus, the derivative of the function is -10t + 3 - 1/t Log t.

8. To find the derivative of the given expression, apply the chain rule. The derivative of 6In5x is 6/x In 5, and the derivative of -2x^2 is -4x. Thus, the derivative of the function is 6/x In 5 - 4x.

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The statement if False.

A family budget is a game plan for your family's money. Your plan identifies where and how your money comes and goes by focusing on income and expenses. Importantly, it also reflects your family's goals and values by how you spend and save.

Given statement,

You overhear a friend discuss how Americans are frivolous in their spending and waste lots of money.

The most expensive items in American spending are needs like housing, taxes, and transportation, while wants like wants are the least expensive.

In America, besides taxes three largest cost categories are households, transportation and food.

A family's transportation costs are $1,847 more than its food expenses.

Typically, needs make up 80% of expenses rather than wants.

Housing, fees, and medical expenses, continuing to be the same each month. The cost of clothing, entertainment, food, and transportation changes every month.

Hence,  we can conclude that, Americans don't necessarily waste money on wants because they have to spend on needs before wants, which results in a lot of spending on things like housing and taxes.

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Answer:

For the first question, To determine the probability that a student studied for exactly 1 hour, we can use the formula:

P(X=1) = (number of students who studied for 1 hour) / (total number of students)

In this case,

P(X=1) = 8/30 = 0.27 (rounded to two decimal places)

So the probability that a student studied for exactly 1 hour is 0.27

For the second question, to determine the experimental probability of drawing a heart, we can use the formula:

P(Heart) = (number of times a heart is drawn) / (total number of draws)

In this case,

P(Heart) = 7/20 = 0.35

So the experimental probability of drawing a heart is 0.35

Therefore, the option 3 is the answer.

Step-by-step explanation:

The value of  ∛64 will be equal to 4.

In mathematics, a number r is said to be the nth root of a number x if it is raised to the power n. where n is a positive integer, also referred to as the root's degree. Degree 3 roots are referred to as cube roots, while degree 2 roots are known as square roots.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

The radical expression will be solved as,

∛64 = ∛ ( 4 x  4 x 4 )

∛64 = ∛ ( 4³ )

∛64 = 4

Hence, the solution for ∛64 will be 4.

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On solving the provided question we can say that the given equation yield solution as x=27.4 and y= -5/8.

An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to explain the connection between two sentences on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.

On solving simultaneously,

x=27.4

y= -5/8

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One equation that represents the total value of the coins in cents is:

0.25x + 0.10y = 995

where x is the number of quarters and y is the number of dimes.

Another equation that represents the total number of coins is:

x + y = 47

These two equations make up a system of equations that can be used to find the number of quarters and dimes that add up to $9.95 and have a total of 47 coins.

Answer:

[tex]\begin{cases}x+y=47\\0.25x+0.1y=9.95\end{cases}[/tex]

Step-by-step explanation:

To solve this problem, we can create and solve a system of equations.

Define the variables:

Values of the coins:

Given there are a total of 47 coins:

[tex]x+y=47[/tex]

Given there is a total of $9.95 in quarters and dimes:

[tex]0.25x+0.1y=9.95[/tex]

Therefore, the system of equations that represents the problem is:

[tex]\begin{cases}x+y=47\\0.25x+0.1y=9.95\end{cases}[/tex]

To solve the system of equations, rewrite the first equation to isolate y:

[tex]y=47-x[/tex]

Substitute this into the second equation to eliminate y:

[tex]0.25x+0.1(47-x)=9.95[/tex]

Solve the equation for x to find the number of quarters:

[tex]\begin{aligned}0.25x+0.1(47-x)&=9.95\\0.25x+4.7-0.1x&=9.95\\0.15x+4.7&=9.95\\0.15x&=5.25\\x&=35\end{aligned}[/tex]

Therefore, there are 35 quarters.

Substitute the found value of x into the first equation and solve for y:

[tex]\begin{aligned}35+y&=47\\y&=12\end{aligned}[/tex]

Therefore, there are 12 dimes.

There are three formulas that can be used to solve quadratic equations in class 10.

The first formula is the Quadratic Formula, which is used to solve any quadratic equation of the form ax² + bx + c = 0. This formula is used to find the two values of x that are the solutions to the equation. The second formula is the Completing the Square formula, which is used to solve any quadratic equation of the form ax² + bx + c = 0. This formula is used to rewrite the equation in the form (x + h)² = p , and then solve for the value of x. The third formula is the Factorization Method, which is used to solve any quadratic equation of the form ax² + bx + c = 0. This method is used to factor the equation into two linear factors, and then use either the Quadratic Formula or the Completing the Square formula to find the two values of x that are the solutions to the equation. All three of these formulas can be used to solve any quadratic equation of the form ax² + bx + c = 0.

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Both expressions are equivalent

What is the expression in mathematics?

In mathematics, an expression is a combination of numbers, variables, and mathematical operations that can be evaluated to a specific value or result. Expressions can be simple or complex, and they can be used to represent mathematical relationships or to perform calculations.

Let c represent the number of ceramic pottery pieces in each box, and let p represent the number of handpainted plates in each box.

To find the total number of ceramic pottery pieces that Solana bought, we can use the following expression:

c* (50 + 58) = c*108

This expression shows that the total number of ceramic pottery pieces is equal to the number of ceramic pottery pieces in each box multiplied by the total number of boxes of ceramic pottery that Solana bought.

Alternatively, we can use the following expression:

c50 + c58 = 108*c

This expression shows that the total number of ceramic pottery pieces is equal to the product of the number of ceramic pottery pieces in each box and the number of boxes Solana bought on Monday plus the product of the number of ceramic pottery pieces in each box and the number of boxes Solana bought on Thursday.

Similarly, to find the total number of handpainted plates that Solana bought, we can use the following expression:

p* (40 + 46) = p*86

This expression shows that the total number of handpainted plates is equal to the number of handpainted plates in each box multiplied by the total number of boxes of handpainted plates that Solana bought.

Alternatively, we can use the following expression:

p40 + p46 = 86*p

This expression shows that the total number of handpainted plates is equal to the product of the number of handpainted plates in each box and the number of boxes Solana bought on Monday plus the product of the number of handpainted plates in each box and the number of boxes Solana bought on Thursday.

Both expressions are equivalent, as they provide the same value for the total number of ceramic pottery pieces and handpainted plates that Solana bought.

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Answer:

5•8-(4+2)=34

Step-by-step explanation:

because you need to change the order of operation

Answer:

slope = [tex]\frac{7}{3}[/tex]

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (5, 2 ) and (x₂, y₂ ) = (8, 9 )

m = [tex]\frac{9-2}{8-5}[/tex] = [tex]\frac{7}{3}[/tex]

Answer:

Step-by-step explanation:

-5/12 = 5/-12

yes

The stakeholders in a stakeholder analysis can include both internal stakeholders and external stakeholders.

Stakeholders are a group that expresses interest in the firm and whose activities, whether professional or personal, have an impact on its operations. The process of identifying and evaluating the significance of the important individuals or groups that are impacted by or have the potential to influence a project's or decision's result is known as stakeholder analysis.

A stakeholder analysis may include both internal and external stakeholders, such as community members, customers, suppliers, regulators, and government organisations. Internal stakeholders in a stakeholder study take into account employees and management. Depending on the project's nature, different parties may be involved, and the list of stakeholders may need to be updated as the project moves along.

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y + x² - 4 is already in simplest form and the value of 6/5 ÷ 1/7 is 8.4.

A fraction with a reasonably prime numerator and denominator is the simplest form. It denotes that the fraction's numerator (upper or top portion) and denominator (lower or bottom portion) share no common component other than 1. A fraction is a value that denotes a portion of a larger whole.

A fraction's simplest form is also known as its reduced form. 3/4 is the simplest form of a fraction with a common component of one, for example. However, because 2/4 can be reduced further and expressed as 1/2, it is not the simplest form. In this case, 1/2 and 2/4 are equivalent fractions.

We have given two question

y + x² - 4 simplify and 6/5 ÷ 1/7 = ?

Lets simplify

y + x² - 4 is already in simplest form

For

6/5 ÷ 1/7

= 6/5 × 7/1

= 42/5

= 8.4

Thus, y + x² - 4 is already in simplest form and the value of 6/5 ÷ 1/7 is 8.4.

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The expression that represents the perimeter of the triangle is 3x

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

Side length = x

Triangle type = equilateral triangle

The perimeter of an equilateral triangle can be calculated as

P = 3 * side length

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

P = 3 * x

So, we have

P =3x

Hence, the perimeter expression is 3x

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The area of the triangle would be 49 cm².

Area of a triangle = (length x breadth)/2

Area of the triangle = (14cm x 7cm)/2

Area of the triangle = (98cm)/2

Area of the triangle = 49 cm²

The area of a triangle can be calculated by using the formula (length x breadth)/2. In this scenario, the length and breadth of the triangle is 14cm and 7cm respectively. Therefore, the area of the triangle can be calculated by using the formula. The area of the triangle would be (14cm x 7cm)/2 = (98cm)/2 = 49 cm². Hence, the area of the triangle with the given length and breadth would be 49 cm².

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