How many floors have exactly 24 offices?

The binomial theorem says

[tex]\displaystyle (a + b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k \\\\(a+b)^n= \binom n0 a^n + \binom n1 a^{n-1}b + \cdots + \binom n{n-1} ab^{n-1} + \binom nn b^n[/tex]

where

[tex]\dbinom nk = \dfrac{n!}{k!(n-k)!}[/tex]

is the so-called binomial coefficient. The binomial coefficients follow a neat pattern called Pascal's triangle:

1

1   1

1   2   1

1   3   3   1

1   4   6   4   1

where the n-th row, starting with n = 0, lists the coefficient of the k-th term in the sum (with 0 ≤ k ≤ n). For example,

[tex]\displaystyle n=0 \implies (a + b)^0 = \sum_{k=0}^0 \binom nk a^{0-k} b^k = \binom00 a^{0-0} b^0 = \underline{1}[/tex]

[tex]\displaystyle n=1 \implies (a + b)^1 = \sum_{k=0}^1 \binom 1k a^{1-k} b^k = \binom10 a^{1-0} b^0 + \binom11 a^{1-1}b^1 = \underline{1}a + \underline{1}b[/tex]

[tex]\displaystyle n=2 \implies (a + b)^2 = \sum_{k=0}^2 \binom 2k a^{2-k} b^k = \binom20a^{2-0}b^0+\binom21a^{2-1}b^1+\binom22a^{2-2}b^2 = \underline{1}a^2 + \underline{2}ab+\underline{1}b^2[/tex]

Each row starts and ends with 1, and every coefficient in between is obtained by adding the coefficients directly above and to the left. The next row, for instance, would be

1   1 + 4   4 + 6   6 + 4   4 + 1   1

or

1   5   10   10   5   1

which is to say,

[tex]\displaystyle (a + b)^5 = \underline{1}a^5+\underline{5}a^4b+\underline{10}a^3b^2+\underline{10}a^2b^3+\underline{5}ab^4+\underline{1}b^5[/tex]

In your case, we have (d - 4b)², so we take a, b, and n above with d, -4b, and 2, respectively:

[tex]\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k d^{2-k} (-4b)^k[/tex]

[tex]\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k (-4)^k d^{2-k} b^k[/tex]

[tex]\displaystyle (d - 4b)^2 = \binom20 (-4)^0 d^{2-0} b^0 + \binom21 (-4)^1 d^{2-1} b^1 + \binom22 (-4)^2 d^{2-2} b^2[/tex]

[tex]\displaystyle (d - 4b)^2 = 1\cdot1\cdot d^2\cdot1 + 2 \cdot (-4) d \cdot b + 1 \cdot 16 \cdot 1 \cdot b^2[/tex]

[tex]\boxed{(d - 4b)^2 = d^2 - 8bd + 16b^2}[/tex]

Find the difference:

17 - (-10) = 27

So the distance between 17 and -10 is 27.

Answer: 27

Step-by-step explanation:

Answer:

the answer is 300

Step-by-step explanation:

30x10=300

If [tex] \: R = 20[/tex]

Then, [tex] \: 10r = 10 \times R \\ = 10 \times 30 \\ = 300[/tex]

Answer:

5

Step-by-step explanation:

3/5 times 7 is 4 1/5 and you cant buy 1/5 of a can so you have to round up to 5

Answer:

so first you add something and then subtract and then divide then you er butf then youcan see toplod ej

Answer:

SEE attched pic

Step-by-step explanation:

Download symbolab

Answer: c

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

Answer:

C. 3, because |-3| = 3

Hope you could get an idea from here.

Doubt clarification - use comment section.

Answer: -8x^2 -2x+13

Step-by-step explanation:

group liked terms

-6x^2-2x^2-2x+5+8

Add numbers 5+8=13

-6^2-2x^2-2x+13

add similiar elements -6x^2-2x^2=-8x^2

-8x^2 -2x+13

Answer:

the answer is C

Step-by-step explanation:

40,000/.80 = 50,000

50,000 x .10 = 5,000

50,000 - 5,000 = 45,000

45,000 x .06 =2700

45,000 + 2,700 = 47,700

Answer:

1. 37.5

2. 8

Step-by-step explanation:

1. the smaller triangle and the bigger triangle are similar, so the sides are also similar.

this means that 16/46 = 20/(x+20)

then, just solve that for x

16/46 = 20/(x+20)

divide both sides by 4

4/46 = 5/(x+20)

multiply both sides by (x + 20)

(x+20) * 4/46 = 5

multiply both sides by 46

4x + 80 = 230

subtract 80

4x = 150

divide 4

x = 37.5

2. do the same thing ( still similar triangles)

the equation would be (3x + 1)/10 = (7x - 1)/22

solve

22(3x + 1) = 10(7x - 1)

66x + 22 = 70x - 10

32 = 4x

x = 8

Julian factored the expression of 2x^4 + 2x³ - x²- x, the step 2 ( x (2x² (x + 1)) - 1(x − 1)) is wrong statement.

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

We have been given an expression as;

2[tex]x^{4}[/tex] + 2x³ - x²- x

Step 1;

x(2x³ + 2x²- x − 1)

Step 2;

x (2x² (x + 1)) - 1(x − 1)

Step 3;

x²(2x²-1) (x + 1)(x - 1)

Hence, Julian incorrectly factored -1 from the second group of terms.

we can write the correct factored as;

x (2x²- 1))(x + 1)

Therefore, Julian factored the expression of 2[tex]x^{4}[/tex] + 2x³ - x²- x, the step 2 ( x (2x² (x + 1)) - 1(x − 1)) is wrong statement.

To know more about an expression follow;

brainly.com/question/19876186

#SPJ5

Answer:

  (250, 150)

Step-by-step explanation:

The acreage cannot be negative, so choices A and C are eliminated. The sum of acreages cannot exceed 400, so choice D is eliminated.

Choice B has the allotted acreages total exactly 400, which is within the farmer's limit.

  (250, 150) is a viable solution

Answer:

They sold 111 Hamburgers

Step-by-step explanation:

Turn this into an equation: first, take the amount of hamburgers and call it 3x due to it being 3 times more than the amount of cheeseburgers, which we will can call x.

So, X+3X=444 is what you need to solve this. This is the same as 4x=444

So, x=111

Now, Input 111 for the final answer; 333 cheeseburgers and 111 hamburgers

Answer:

7 square feet

Step-by-step explanation:

There are seven squares, one is 1 square foot

7x1=7

Answer:

(-2,1)(-1,1)(0,1)(1,1) or y=2 x=[-2,1]

Step-by-step explanation:

Brainliest Please

Using permutations and the probability concept, it is found that there is a 0.0417 = 4.17% probability that Nia is chosen as president.

Permutation formula:

The number of possible permutations of x elements from a set of n elements is given by:

[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]

Desired outcomes:

[tex]D = P_{23,3} = \frac{23!}{20!} = 10626[/tex]

Total outcomes:

4 students from a set of 24, hence:

[tex]T = P_{24,4} = \frac{24!}{20!} = 255024[/tex]

Then:

[tex]p = \frac{D}{T} = \frac{10626}{255024} = 0.0417[/tex]

0.0417 = 4.17% probability that Nia is chosen as president.

To learn more about probabilities, you can take a look at https://brainly.com/question/24437717

Answer:The answer is 23 P 3/ 24 P 4

Step-by-step explanation:

Answer:

27 in²

Step-by-step explanation:

The area (A) is calculated as

A = πr² ( r is the radius ) , then

A = π × 3² = 3 × 9 = 27 in²

Answer:

Option B is the correct answer

Step-by-step explanation:

Complimentary angles mean sum of angles is [tex]90^{0}[/tex]

Let the smaller angle be [tex]x[/tex]

Other angle is [tex]90-x[/tex]

We have

Two complementary angles are in the ratio [tex]3: 2[/tex]

[tex]\frac{90-x}{x} =\frac{3}{2} \\\\180-2x=3x\\\\5x=180\\\\x=36[/tex]

Smaller angle is [tex]36^0[/tex]

Option B is the correct answer

Answer:

Step-by-step explanation:

The coefficients of x and y do not have any "nice" relations, so doing this by elimination is just "brute force" with no particular subtlety.

After multiplying by appropriate numbers, you want the coefficient of one of the variables in one equation to be the opposite of the coefficient of the same variable in the other equation. Here, we can get that by multiplying the first equation by 7 and the second equation by 2.

  7(2x +3y = -19)   ⇒   14x +21y = -133

  2(-7x +2y = -21)   ⇒   -14x +4y = -42

Adding these equations with opposite x-coefficients eliminates the x-terms. (That is how the "elimination" method gets its name.)

  (14x +21y) +(-14x +4y) = (-133) +(-42)

  25y = -175 . . . . . . . . simplify

  y = -7 . . . . . . . . . . . divide by 25

Substituting this value into the first equation gives ...

  2x +3(-7) = -19

  2x = 2 . . . . . . . . . add 21

  x = 1 . . . . . . . . . divide by 2

The solution is x = 1, y = -7.

_____

Additional comment

The multipliers used in this process are essentially the coefficients of one of the variables, with one of them negated. Here, we chose the x-coefficients, because negating one of them gives two positive multipliers. We judge it easier to do the arithmetic using positive numbers.

Had we chosen to use the y-coefficients (3 and 2) for our multipliers, we would probably multiply the first equation by 2, and the second equation by -3. That way, the remaining x-coefficient would be positive.

Planning ahead is helpful, but not essential. It can help to avoid errors and make the work easier to do.

Answer:

1728

Step-by-step explanation:

27

64

-----------

108

1620

1728

Answer:

1728

Step-by-step explanation:

27•64= 0108+162 =1728

Answer:

  (x, y) = (6, 2)

Step-by-step explanation:

You solve for x and y by writing equations that reflect the relation applicable to the given geometry.

The diagonals of a parallelogram bisect each other. That means the halves of each diagonal are equal.

  2x -4 = 4y . . . . . . down diagonal

  x +4 = 7y -4 . . . . . up diagonal

__

We can subtract the first equation from 2 times the second equation to eliminate the x-variable.

  2(x +4) -(2x -4) = 2(7y -4) -(4y)

  2x +8 -2x +4 = 14y -8 -4y . . . . . . eliminate parentheses

  12 = 10y -8 . . . . . . . . . . . . . . . . . collect terms

  20 = 10y . . . . . . . . . . . . . . . . . . add 8

  2 = y . . . . . . . . . . . . . . . . . . . . divide by 10

Substituting into the second equation gives ...

  x +4 = 7(2) -4

  x = 6 . . . . . . . . . . . . subtract 4

The values of x and y are (x, y) = (6, 2).

_____

The lengths of the diagonals are 20 and 16.

Answer:

the image was not clear....hope the values are correct

Answer is in the attached file.