Consider two natural numbers n,k and p where 0≤k≤p≤n a. Prove that:
P(n+1,k)=P(n,k)+kP(n,k−1) b. Prove that: C(n,k)C(n−k,p−k)=C(p,k)C(n,p)

If a store sells two types of shirts. The number of short-sleeved shirts that were sold than long-sleeved shirts is 24 shirts.

Number of short-sleeved shirts sold =S

Number of long-sleeved shirts sold = L

we know that:

s + l = 48 (equation 1) -- Total number of shirts sold

12s + 16l = 624 (equation 2) -- Total cost of shirts sold

To solve for the number of short-sleeved shirts sold, we can use equation 1 to express "l" in terms of "s":

l = 48 - s

Substituting this into equation 2

12s + 16(48 - s) = 624

Simplifying and solving for "s"

12s + 768 - 16s = 624

-4s = -144

s = 36

Therefore, 36 short-sleeved shirts were sold.

To find the number of long-sleeved shirts sold, we can use equation 1 again:

s + l = 48

36 + l = 48

l = 12

Therefore, 12 long-sleeved shirts were sold.

How many MORE short-sleeved shirts were sold than long-sleeved shirts,:

36 - 12 = 24

So, 24 MORE short-sleeved shirts were sold than long-sleeved shirts.

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Answer:

Step-by-step explanation: m miles

Based on the coordinate plane, the ordered pair for the post office is (9, 9).

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.

By critically observing the coordinate plane shown in the image attached below, we can logically deduce that the coordinates of the post office would be located in quadrant I at point (9, 9).

Similarly, the the coordinates of the library is also located in quadrant I and it is represented by the ordered pair (6, 2).

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Answer: just do 30 - 13 which = 17

Step-by-step explanation:

subtraction above

The most specific category that this shape can be sorted into is "Triangle."

What is the triangle?

A triangle is a three-sided polygon, which has three vertices. The three sides are connected with each other end to end at a point, which forms the angles of the triangle. The sum of all three angles of the triangle is equal to 180 degrees.

The Venn diagram shows the relationships between different types of triangles.

The Scalene circle represents all the triangles that have no equal sides.

The Isosceles circle represents all the triangles that have two equal sides.

The Equilateral circle represents all the triangles that have three equal sides.

The triangle with a tick mark on each side does not provide enough information to determine whether it is a scalene, isosceles, or equilateral triangle.

Therefore, the most specific category that this shape can be sorted into is "Triangle."

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Answer: -7.6n + 11.1

Step-by-step explanation:

We will simplify the given expression by combining like-terms.

   Given:

3.1 - 4.6n - 3n + 8

  Subtract like-terms:

3.1 - 7.6n + 8

   Add like-terms:

11.1 - 7.6n

Answer:

[tex] \sf \: -7.6n + 11.1 [/tex]

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The expression is,

→ 3.1 - 4.6n - 3n + 8

Let's simplify the expression,

→ 3.1 - 4.6n - 3n + 8

→ -4.6n - 3n + 3.1 + 8

→ (-4.6n - 3n) + (3.1 + 8)

→ (-7.6n) + (11.1)

→ -7.6n + 11.1

Hence, the answer is -7.6n + 11.1.

The measure of angle ABD as required to be determined from the task content is; 32°.

As evident in the task content;

m<ABD + m<CBD = m<ABC = 90°.

This follows from the fact that the angle ABC is a right angle.

4x - 4 + 2x + 40 = 90

6x = 54

x = 54 / 6 = 9

Ultimately, the measure of angle ABD is; 4(9) - 4 = 36 - 4 = 32.

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Answer:

QU is 21, and I *think* that VS would be 8.5.

Answer:

QV = 21 , VS = 8.5

Step-by-step explanation:

QU and RS are medians of Δ PQR

the point V where the medians intersect is the centroid.

on each median the distance from the vertex to the centroid is twice as long as the distance from the centroid to the midpoint.

then

VU = [tex]\frac{1}{2}[/tex] QV = [tex]\frac{1}{2}[/tex] × 14 = 7

so

QU = QV + VU = 14 + 7 = 21

and

VS = [tex]\frac{1}{2}[/tex] RV = [tex]\frac{1}{2}[/tex] × 17 = 8.5

Answer:

Step-by-step explanation:

a. To find the area of the sector, we can use the formula:

A = (θ/360)πr^2

where A is the area of the sector, θ is the central angle in degrees, r is the radius of the circle, and π is the constant pi.

In this case, the radius is 4 cm and the central angle is 45 degrees. Substituting these values into the formula, we get:

A = (45/360)π(4^2)

A = (1/8)π(16)

A = 2π

Therefore, the area of the sector is 2π square cm.

b. To find the area of the segment of a circle, we need to subtract the area of the triangle formed by the two radii and the chord from the area of the sector.

The central angle of the sector is 45 degrees, so the angle between the chord and one of the radii is 22.5 degrees. We can use trigonometry to find the length of the chord:

cos(22.5) = adjacent/hypotenuse

cos(22.5) = x/4

x = 4cos(22.5)

So the length of the chord is approximately 3.54 cm (rounded to two decimal places).

The area of the triangle can be found using the formula:

A = (1/2)bh

where b is the length of the base (which is the chord) and h is the height (which is the distance from the midpoint of the chord to the center of the circle). The height is equal to the radius minus half the length of the chord:

h = 4 - (3.54/2)

h = 1.23 (rounded to two decimal places)

Substituting the values of b and h, we get:

A = (1/2)(3.54)(1.23)

A = 2.17 (rounded to two decimal places)

So the area of the triangle is approximately 2.17 square cm.

Finally, we can find the area of the segment by subtracting the area of the triangle from the area of the sector:

Area of segment = Area of sector - Area of triangle

Area of segment = 2π - 2.17

Area of segment = 0.85 (rounded to two decimal places)

Therefore, the area of the segment of the circle is approximately 0.85 square cm.

Answer: Let's use algebra to check if Tim is correct.

Let x be the original price of the jeans.

When the jeans are marked down by 20%, the new price becomes:

x - 0.2x = 0.8x

Tim claims that if we increase the new price by 20%, we will get the original price:

0.8x + 0.2(0.8x) = x

Simplifying the left-hand side:

0.8x + 0.16x = x

0.96x = x

This is not true for all values of x, which means that Tim's claim is not correct.

Let's use the given information that the original price is $200 to find out what the new price is after the 20% discount:

New price = Original price - Discount

New price = $200 - 20% of $200

New price = $200 - $40

New price = $160

If we increase the new price of $160 by 20%, we get:

New price + 20% of new price

$160 + 20% of $160

$160 + $32

$192

This is not equal to the original price of $200, so we can conclude that Tim's claim is incorrect.

Answer:

25&36

Step-by-step explanation:

25 is a square number and so is 36 and there product is 900. They are also consecutive.

Just assemble the square numbers from the least i.e 4 and try solving as asked to see if it gives 900.

Thus you'll land on 25&36

Answer:

5² and 6²

Step-by-step explanation:

Use trial and error

Start from the product of 1² and 2²

1×2²

1×4=4

4<900 (incorrect)

2²×3²

4×9=36

36<900(incorrect)

3²×4²

9×16=144

144<900(incorrect)

4²×5²

16×25=400

400<900(incorrect) *but close

5²×6²

25×36 =900

900=900(correct)

: . 5² and 6² are the two consecutive square numbers.

The given equation x² + 12x − 1  can be written in the form of algebraic identity (x + 6)² -37

Algebraic identities are equations in algebra that hold regardless of the value of each of their variables. The factorization of polynomials makes use of algebraic identities. On both sides of the equation, they have variables and constants.

Write x² + 12x − 1 in the form (x + a)² + b, where a and b are constants.

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added to a variable before being multiplied by it.

[tex]The \ given \ equation\ is\ \\\\x^2+12x-1\\let , \ x^2+12x=1\\\\add \ both\ side \ \frac{12^2}{4}\\\\x^2+12x+\frac{144}{4}=1+\frac{144}{4}\\\\x^2+12x+36=1+36\\x^2+2.6.x+6^2-37\\\\(x-6)^2-37\\\\[/tex]

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4) Let the polynomial function of degree 4 that has roots 3, 2, 2+ i as its roots be p(x).

So, the required polynomial function is:

p(x)=(x−3)(x−2)(x−(2+i))(x−(2−i))

= (x−3)(x−2)(x^2−(2+i)x−(2−i)x+(2−i)(2+i))

= (x−3)(x−2)(x^2−2x−ix+ix+4)

= (x−3)(x−2)(x^2−2x+4)

= x^4−9x^3+28x^2−36x+24

Thus, the polynomial function of degree 4 that has real coefficients and has 3, 2 and 2+ i as some of its roots is x^4−9x^3+28x^2−36x+24.

Given: x^3−2x^2+36x−72=0 and 6i is a zero of this polynomial function.

So, we can write it as:

x^3−2x^2+36x−72= (x−6i)(x−(−6i))(x−6)

= (x−6i)(x+6i)(x−6)

As this polynomial function has real coefficients, so the imaginary roots occur in conjugate pairs.

Thus, the remaining zeros are:

−6i (as 6i is a zero, so −6i is also a zero due to conjugate pair of complex roots) and 6 (as 6i is one factor of the polynomial function, so x−6i will be its conjugate factor, which will be x+6i. And, the remaining factor will be x−6)

Therefore, the remaining zeros of x^3−2x^2+36x−72 polynomial function are −6i and 6.

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Answer:

a= 528 squared centimeters

Step-by-step explanation:

Area of trapezoid is A=[(a+b)/2]*h

Base 1 is 32 cm

height is 12

since the height is also same as the bottom part as told by the line, do 32+12+12 to get...

56 cm for base 2

Sooo put this to use

a=[(32+56)/2]*12

a=[(88)/2]*12

a=[44]*12

a=528 :)

Answer:

Area 528cm²

That's the answer to your question.

The probability of two people on the same team having the same birthday is: P(A) = 1 / 16

The probability of two people having the same birthday in a group of 32 people is 1/365. This is because there are 365 possible days that a person can have a birthday, and the probability of two people having the same birthday is 1/365.

In this case, there are 16 teams of 2 students each, and the probability of two people on the same team having the same birthday is 1/365.

To calculate the probability of this event occurring, we can use the formula:

P(A) = n(A) / n(S)

Where P(A) is the probability of the event occurring, n(A) is the number of favorable outcomes, and n(S) is the total number of possible outcomes.

In this case, the number of favorable outcomes is 1, since there is only one team that has two people with the same birthday. The total number of possible outcomes is 16, since there are 16 teams.

Therefore, the probability of two people on the same team having the same birthday is:

P(A) = 1 / 16

So, the probability of this event occurring is 1/16.

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The sοlutiοn tο the equatiοn is x=7.

An equatiοn is an expressiοn that uses mathematical symbοls tο express the relatiοnship between twο οr mοre variables. Equatiοns are used tο describe physical laws, mοdel real-wοrld prοblems, and sοlve mathematical prοblems. Equatiοns can be written in a variety οf fοrms, frοm simple linear equatiοns tο cοmplex nοnlinear equatiοns. Equatiοns can alsο be used tο determine the prοperties οf certain functiοns and tο evaluate integrals.

Sοlving fοr x,

−10(−x+1)+10+10(x+3)=0

−10x+10−10x+10+100+30=0

−20x+140=0

20x=140

x=7

This equatiοn is an example οf a linear equatiοn. Linear equatiοns are equatiοns that invοlve οnly οne variable and can be represented in the fοrm ax + b = 0, where x is the variable and a and b are cοnstants. Linear equatiοns are useful fοr understanding the relatiοnship between different variables and can be used tο sοlve real-wοrld prοblems. In this equatiοn, the variable x is the unknοwn value that we are trying tο sοlve fοr. By rearranging the equatiοn and applying the apprοpriate algebraic οperatiοns, we were able tο sοlve fοr x. This is an example οf hοw linear equatiοns can be used tο sοlve real-wοrld prοblems.

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Solve for x in given expression.

10 + 10(x + 3) = −x −10(−x + 1)

0.3406 (approx)

Given that 60% of all Americans live in cities with population greater than 100,000 people. If 47 Americans are randomly selected, we need to find the probability thata. Exactly 25 of them live in cities with population greater than 100,000 people.b. At most 26 of them live in cities with population greater than 100,000 people.c. At least 30 of them live in cities with population greater than 100,000 people.d. Between 23 and 28 (including 23 and 28) of them live in cities with population greater than 100,000 people.Probability is defined as the ratio of the favorable outcomes to the total outcomes of an event. The formula to calculate probability is given by;Probability = Number of favorable outcomes / Total number of outcomesa. Exactly 25 of them live in cities with a population greater than 100,000 people.Probability of exactly 25 Americans living in cities with population > 100,000 people is given by the probability mass function of the binomial distribution.P( X = 25) = 47 C 25 * (0.6)25 * (0.4)22= 0.1213 (approx)b. At most 26 of them live in cities with population greater than 100,000 people.We need to find the probability of at most 26 Americans living in cities with population greater than 100,000 people. This means the number of Americans living in cities with population greater than 100,000 is 0, 1, 2, ..., 26.P(X ≤ 26) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 26)P(X ≤ 26) = ΣP(X = r)Where r varies from 0 to 26P(X ≤ 26) = Σ (47 C r * (0.6)r * (0.4)47-r )r = 0 to 26= 0.6413 (approx)c. At least 30 of them live in cities with population greater than 100,000 people.We need to find the probability of at least 30 Americans living in cities with population greater than 100,000 people. This means the number of Americans living in cities with population greater than 100,000 is 30, 31, 32, ..., 47.P(X ≥ 30) = P(X = 30) + P(X = 31) + P(X = 32) + ... + P(X = 47)P(X ≥ 30) = ΣP(X = r)Where r varies from 30 to 47P(X ≥ 30) = Σ (47 C r * (0.6)r * (0.4)47-r )r = 30 to 47= 0.0031 (approx)d. Between 23 and 28 (including 23 and 28) of them live in cities with population greater than 100,000 people.We need to find the probability of the number of Americans living in cities with population greater than 100,000 is between 23 and 28 (both inclusive).P(23 ≤ X ≤ 28) = P(X = 23) + P(X = 24) + ... + P(X = 28)P(23 ≤ X ≤ 28) = ΣP(X = r)Where r varies from 23 to 28= Σ (47 C r * (0.6)r * (0.4)47-r )r = 23 to 28= 0.3406 (approx)Hence, the required probabilities are,a. P(X = 25) = 0.1213 (approx)b. P(X ≤ 26) = 0.6413 (approx)c. P(X ≥ 30) = 0.0031 (approx)d. P(23 ≤ X ≤ 28) = 0.3406 (approx)

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Answer: x=10/9

Step-by-step explanation:

around 13 jars of paint will the farmer need to buy in order to paint the entire exterior of the silo.

To find the number of jars the rancher required to get, you really want to know the surface area of both the cone and cylinder.

The method for finding the SA of the cone would be area =[tex]3.14 x r^2 + 3.14 x r x sqrt(r^2 + h^2)[/tex].

SA= 1976.0617791 [tex]m^2[/tex].

The method for finding the SA of the cylinder under the cone would be 3.14 x d x (d/2 + h).

A = 3097.6103564 [tex]m^2[/tex]

Then, at that point, you need to add 1976.0617791+3097.6103564 which gives you 5,073.8. (5,073 is the total surface area of the storehouse)

Then, the partition that by 400.

5,073.8/400 = 12.7

He would have to purchase around 13 jars of paint to have the option to paint the whole storehouse.

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Answer:

13

Step-by-step explanation:

To find the distance between two points in a coordinate plane, we can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Using this formula, we can find the distance between the points (-3, 6) and (-8, -6) as follows:

d = sqrt((-8 - (-3))^2 + (-6 - 6)^2)

= sqrt((-5)^2 + (-12)^2)

= sqrt(25 + 144)

= sqrt(169)

= 13

Therefore, the distance between the two points in simplest radical form is 13.

ECDF=1/100

The value of the empirical cumulative distribution function (ECDF) at the smallest observation in a sample of 100 observations is 1/100.What is the empirical cumulative distribution function (ECDF)?The empirical cumulative distribution function (ECDF) is a graph that depicts the distribution of a given data set. It is used to illustrate the proportion or percentage of data points that fall below a particular value in the distribution. The ECDF can be used to construct a distribution function for data that do not have an existing theoretical distribution.The ECDF value at the smallest observation in a sample of 100 observations is 1/100. This is because the ECDF takes on a value of 0 for all values that are smaller than the smallest observation. Since there are 100 observations in the sample, each observation has a weight of 1/100. Thus, the ECDF value at the smallest observation is 1/100.

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for (d), the x value has changed, as the (-1) in the series acts as a multiplier and flips the convergence of the original series. This means that when x = -4, the original series converges, but when x = -7, the series in (d) converges. Therefore, the series in (d) converges.

(a) Diverges

(b) Diverges

(c) Converges

(d) Converges: The original series converges when x = -4 and diverges when x = 6. For (a), (b), and (c), the x value remains the same as in the original series, so the convergence or divergence of the series is the same as the original series. However, for (d) the x value has changed. The (-1) in the series acts as a multiplier and flips the convergence of the original series, so the series converges.

The convergence or divergence of a series is determined by the value of x in the series. In this particular case, when x = -4 the original series converges, and when x = 6 it diverges. For (a), (b), and (c), the x value is the same as in the original series, so the convergence or divergence of the series is the same as the original series. However, for (d), the x value has changed, as the (-1) in the series acts as a multiplier and flips the convergence of the original series. This means that when x = -4, the original series converges, but when x = -7, the series in (d) converges. Therefore, the series in (d) converges.

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Answer:

It's very difficult

maby

The value of a is -9

When expressing a polynomial in its standard form, the greatest degree of terms are written first, followed by the next degree, and so on.

[tex]P(x)=4x^5+3x^2+2x+a[/tex]

Question might be asking to find the value of 'a'  at the point (1, 0) on the graph of  [tex]P(x)=4x^5+3x^2+2x+a[/tex]

Substitute the point into the polynomial.  i.e., x=1, y=0

=>  [tex]0=4(1)^5+3(1)^2+2(1)+a[/tex]

=> 0=  4*1 + 3*1+2 +a

=> 0= 4+3+2+a

=> 0=9 +a

=> a= -9

The value of a is -9

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Complete Question:

Find the value of  [tex]P(x)=4x^(5)+3x^(2)+2x+a[/tex]

The new mural dimensions are 267 inches by 273 inches.To find the new dimensions, we must add 33 inches to the original length of 234 inches, giving us a new length of 267 inches.

To calculate the new dimensions of Rhiannon's mural, we must first identify the original dimensions of the mural which are 234 inches by 245 inches. To find the new dimensions, we must add 33 inches to the original length of 234 inches, giving us a new length of 267 inches. We must also add 28 inches to the original width of 245 inches, giving us a new width of 273 inches. Therefore, the new dimensions of Rhiannon's mural are 267 inches by 273 inches.

The complete question is :

A rectangle mural measures 234 inches by 245. Rhiannon creates a new mural that is 33 inches longer. What are the dimensions of Rhiannon's new mural?

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Singular Savings Bank received an initial deposit of $3000. By keeping a percentage of this money in reserve based on the reserve rate, the bank loaned out the rest. Over time, the money loaned out was all deposited back into the bank, and with each additional deposit the bank was able to lend out even more money.

This cycle of lending and depositing continued indefinitely until the initial $3000 had increased to $50,000. This increase in money was possible due to the reserve rate, which allowed the bank to lend out a percentage of the money deposited and to keep a percentage in reserve for themselves.

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The correct statement regarding the translation of the functions f(x) and h(x) is given as follows:

The graph of h is a translation of 4 units left and 7 units down of f(x).

A translation happens when either a figure or a function are moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The meaning of each translation from function f(x) to function h(x) is given as follows:

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