Let $n$ be a positive integer.
(a) There are $n^2$ ordered pairs $(a,b)$ of positive integers, where $1 \le a,$ $b \le n.$ Using a counting argument, show that this number is also equal to
\[n + 2 \binom{n}{2}.\]
(b) There are $n^3$ ordered triples $(a,b,c)$ of positive integers, where $1 \le a,$ $b,$ $c \le n.$ Using a counting argument, show that this number is also equal to
\[n + 3n(n - 1) + 6 \binom{n}{3}.\]
Answers
Alternatively, we can count the number of ordered pairs $(a,b)$ by dividing into cases based on the value of $a.$ For each $a,$ there are $n$ choices for $b$ (namely, $b$ can be any integer between 1 and $n,$ inclusive). Thus, the total number of ordered pairs is $\sum_{a=1}^n n = n^2.$
On the other hand, we can also count the number of ordered pairs $(a,b)$ by first choosing two distinct integers from the set $\{1,2,\ldots,n\}$ and then ordering them. There are $\binom{n}{2}$ ways to choose two distinct integers from the set, and once we have chosen them, there are two ordered pairs corresponding to them (namely, $(a,b)$ and $(b,a)$). Thus, the total number of ordered pairs is $2\binom{n}{2}.$
Since we have counted the same quantity in two different ways, we must have
\[n^2 = 2\binom{n}{2} + n.\]
(b) We can count the number of ordered triples $(a,b,c)$ in three ways.
First, we can simply note that there are $n$ choices for each of $a,$ $b,$ and $c,$ giving a total of $n^3$ ordered triples.
Alternatively, we can count the number of ordered triples $(a,b,c)$ by dividing into cases based on the values of $a$ and $b.$ For any given pair $(a,b),$ there are $n$ choices for $c,$ so the total number of ordered triples is $\sum_{a=1}^n \sum_{b=1}^n n = n^3.$
On the other hand, we can also count the number of ordered triples $(a,b,c)$ by first choosing three distinct integers from the set $\{1,2,\ldots,n\}$ and then ordering them. There are $\binom{n}{3}$ ways to choose three distinct integers from the set, and once we have chosen them, there are $3! = 6$ ordered triples corresponding to them (namely, $(a,b,c),$ $(a,c,b),$ $(b,a,c),$ $(b,c,a),$ $(c,a,b),$ and $(c,b,a)$). Thus, the total number of ordered triples is $6\binom{n}{3}.$
Finally, we can count the number of ordered triples $(a,b,c)$ by dividing into cases based on how many of $a,$ $b,$ and $c$are equal. If all three are equal, there are $n$ choices for each of $a,$ $b,$ and $c,$ giving a total of $n$ ordered triples. If exactly two are equal, there are $3n(n-1)$ choices for $(a,b,c)$ (namely, we can choose the two equal values in $n$ ways, and then choose the distinct value in $n-1$ ways). If all three are distinct, there are $6\binom{n}{3}$ choices for $(a,b,c)$ (as before). Thus, the total number of ordered triples is
\[n + 3n(n-1) + 6\binom{n}{3}.\]
Since we have counted the same quantity in three different ways, we must have
\[n^3 = n + 3n(n-1) + 6\binom{n}{3}.\]
Related Questions
statistics computed for larger random samples are less variable than the statistic computed for smaller random samples
Answers
Statistics computed for larger random samples tend to be less variable compared to statistics computed for smaller random samples.
This statement is based on the concept of the Central Limit Theorem (CLT) in statistics. According to the CLT, as the sample size increases, the distribution of the sample mean approaches a normal distribution regardless of the shape of the population distribution. This means that the variability of the sample mean decreases as the sample size increases.
The variability of a statistic is commonly measured by its standard deviation or variance. When working with larger random samples, the individual observations have less impact on the overall variability of the statistic. As more data points are included in the sample, the effects of outliers or extreme values tend to diminish, resulting in a more stable and less variable estimate.
In practical terms, this implies that estimates or conclusions based on larger random samples are generally considered more reliable and accurate. Researchers and statisticians often strive to obtain larger sample sizes to reduce the variability of their results and increase the precision of their statistical inferences.
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The midpoint of AB is M(3, -1). If the coordinates of A are (5,1), what are the
coordinates of B?
Answers
Given parameters:
Midpoint of AB = M(3, -1)
Coordinates of A = (5,1)
Unknown:
Coordinates of B = ?
Solution:
To find the mid point of any line, we use the expression below;
\(x_{m} = \frac{x_{1} + x_{2} }{2}\) and \(y_{m} = \frac{y_{1} + y_{2} }{2}\)
where \(x_{m}\) and \(y_{m}\) = coordinates of the mid points = 3 and -1
x₁ = 5 and y₁ = 1
x₂ = ? and y₂ = ?
Now let us input the variables and solve,
3 = \(\frac{5 + x_{2} }{2}\) and -1 = \(\frac{1 + y_{2} }{2}\)
5 + x₂ = 6 -2 = 1 + y₂
x₂ = 1 y₂ = -2 -1 = -3
The coordinates of B = 1, -3
Write the expression as a single multiplication sentence without
parentheses.
(75 + 25) × 38
Answers
Hope this helps please give brainliest❤️
Answer:
100×38
Step-by-step explanation:
Add the 75 and the 25, then that is what you multiply with 38
please help i don’t get it :(
Answers
10 x 10 = 100
the area is 100 m^2
poni wants to purchase a birdfeeder that costs $19.95. The sales-tax rate in her city is 812% . What is the total cost of the birdfeeder?
Answers
The total cost of the birdfeeder which poni wants to buy including the sales tax is $21.54.
Given poni wants to purchase a birdfeeder that costs $19.95.
sales tax rate in her city is = 8.12%
total cost of the birdfeeder = ?
The pre-tax price of an item must first be multiplied by the sales tax percentage once it has been converted into a decimal form in order to determine the amount of sales tax that will apply to the purchase. To get the item's final cost, the sales tax must first be computed, and then it must be added to the pre-tax price.
sales tax rate = 8.12% = 0.0812
Round it off to two decimal places.
= 0.08
Now, in order to determine the cost of the sales tax, we must multiply the item's pre-tax price by this amount.
sales tax = 0.08×$19.95
sales tax = $1.596
Since our total is in dollars and cents, round to two decimal places.
sales tax = $1.59
now total cost = pre tax value+sales tax.
= $19.95+1.59
=$21.54
Hence the total cost of the birdfeeder is $21.54.
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How do you find graph of equation?.
Answers
Step-by-step explanation:
You must know the properties of equations.
If the graph is a line, then
the form of the equation will be either
\(ax + by = c\)
or
\(y = mx + b\)
If the graph is shaped like a U( parabola), then our equation will be
\(ax {}^{2} + bx + c\)
Under the consent once removed exception, officers cannot make a warrantless entry to arrest a suspect if consent to enter was given earlier to an undercover officer or informant. Group of answer choices True False
Answers
The statement, "Under consent once removed-exception, officers cannot make a "warrantless-entry" to arrest suspect" is False because officers can make a warrantless entry.
This exception allows law enforcement officers to rely on the consent initially given by the suspect to the undercover officer or informant as a basis for making a warrantless entry to effectuate an arrest.
The rationale behind this exception is that the suspect's consent remains valid even if it was given to someone acting undercover or in an informant capacity.
Therefore, the statement that officers cannot make a warrantless entry under the consent once removed exception is false.
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Use the Side-Splitter Theorem to find x given that NQ is parallel to OP.
Answers
Answer:
x = 7
Step-by-step explanation:
\(\frac{20}{30} =\frac{14}{14+x}\)
\((20)(14+x)=(30)(14)\)
\(280+20x=420\)
\(20x=420-280\)
\(20x=140\)
\(x=140/20=7\)
Hope this helps
ANSWER NOW AND YOU GET 20 and ill answer one of ur questions
Answers
Answer:
300 gallons per hour.
Step-by-step explanation:
It takes 4 of 1/4 hours to make 1 hour. Now multiply gallons per quarter hour by four.
75 x 4 = 300
Consider using a z test to test
H0: p = 0.9.
Determine the P-value in each of the following situations. (Round your answers to four decimal places.)
(a)
Ha: p > 0.9, z = 1.44
(b)
Ha: p < 0.9, z = −2.74
(c)
Ha: p ≠ 0.9, z = −2.74
(d)
Ha: p < 0.9, z = 0.23
Answers
The P-values for the given situations are approximately 0.0749, 0.0030, 0.0059, and 0.4108, respectively.
To determine the P-value in each situation, we need to find the area under the standard normal distribution curve that corresponds to the given z-values.
(a) Ha: p > 0.9, z = 1.44:
The P-value for this situation corresponds to the area to the right of z = 1.44. Using a standard normal distribution table or a calculator, we find the P-value to be approximately 0.0749.
(b) Ha: p < 0.9, z = -2.74:
The P-value for this situation corresponds to the area to the left of z = -2.74. Using a standard normal distribution table or a calculator, we find the P-value to be approximately 0.0030.
(c) Ha: p ≠ 0.9, z = -2.74:
The P-value for this situation corresponds to the area to the left of z = -2.74 (in the left tail) plus the area to the right of z = 2.74 (in the right tail). Using a standard normal distribution table or a calculator, we find the P-value to be approximately 0.0059.
(d) Ha: p < 0.9, z = 0.23:
The P-value for this situation corresponds to the area to the left of z = 0.23. Using a standard normal distribution table or a calculator, we find the P-value to be approximately 0.4108.
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√xx − 2 = 7
a. 51
b. 27
c. 23
Answers
Answer: c=23
Step-by-step explanation: Honest guess
please answer my question
Answers
Answer:
your questions is not complete yet please send full questions and write your question ⁉️ so I will solve your problem
Describe spatial interpolation by inverse distance weighting
method, its equation, parameters and properties.
Answers
Inverse distance weighting (IDW) spatial interpolation is a technique for estimating values at unknown places from nearby known values. The equation for IDW is: Z(x) = Σ [wi * Zi] / Σ wi. The power parameter (p) and the search radius (r) are among the IDW's parameters.
Spatial interpolation by inverse distance weighting (IDW) is a method used to estimate values at unknown locations based on nearby known values. It is commonly used in geostatistics and spatial analysis to fill in missing or unobserved data points in a continuous surface.
The equation for IDW is as follows:
Z(x) = Σ [wi * Zi] / Σ wi
In this equation,
Z(x) represents the estimated value at location x,
Zi represents the known value at location i, and
wi represents the weight assigned to each known value based on its distance from location x.
The parameters of IDW include the power parameter (p) and the search radius (r).
The power parameter determines the influence of each known value on the estimated value at the unknown location. A higher power value gives more weight to the closest points, while a lower power value spreads the influence of nearby points more evenly.
The search radius defines the distance within which neighboring points are considered for interpolation.
IDW has several properties that are important to consider:
1. Inverse relationship: IDW assumes an inverse relationship between distance and influence. Closer points have a greater influence on the estimated value than farther points.
2. Deterministic: IDW provides a deterministic estimate at each unknown location based on the known values within the search radius.
3. Smoothing effect: IDW tends to smooth out abrupt changes in the data. This can be an advantage when dealing with noisy or inconsistent data, but it can also result in the loss of detailed information.
4. Sensitivity to parameter selection: The choice of power parameter and search radius can significantly impact the results of IDW. It is important to select appropriate values based on the characteristics of the data and the desired outcome.
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What value of x makes the
equation frue?
X - 13 = 25
Answers
Answer:
38
Step-by-step explanation:
x-13=25
+13 +13
_________
x=38
Answer:
38
Step-by-step explanation:
38-13=25.
To find this add thirteen to 25
A lilly pond starts with 1 lilly pad and every day the amount doubles. how many lilly pads are in the pond after d days
Answers
After d days, the number of lily pads in the pond can be calculated using the formula 2^d. So, if d is the number of days, then the number of lily pads after d days would be 2^d.
Each day, the number of lily pads doubles. So, on the first day, there is 1 lily pad. On the second day, the number doubles to 2. On the third day, it doubles again to 4, and so on. This doubling pattern continues for d days.
To calculate the number of lily pads after d days, we raise 2 to the power of d (2^d). This is because each day, the number of lily pads doubles, which can be represented as 2^1, 2^2, 2^3, and so on. By substituting the value of d into the equation, we can find the number of lily pads after d days.
For example, if d = 5, then the number of lily pads after 5 days would be 2^5 = 32. This means that there would be 32 lily pads in the pond after 5 days.
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en una tienda de ropa, la semana pasada, 3 pantalones y 2 abrigos costaban 245€. Esta semana estos artículos tienen un descuento del 20% y 5% respectivamente. Si ahora un pantalón y un abrigo cuestan 100,75€, ¿qué costaba cada artículo antes de la rebaja?
Answers
Solving a system of equations we can see that each pair of pants costs €82.94 and each coat costs €26.37
How to find the original cost?
Let's define the variables:
x = cost of a pant
y = cost of a coat.
First, we know that:
3x + 2y = 245
And then there are discounts of 20% and 5%, and the cost of one of each is 100.75, then:
0.8x + 0.95y = 100.75
Then we have a system of equations:
3x + 2y = 245
0.8x + 0.95y = 100.75
We can isolate x on the second equation to get:
x = (100.75 - 0.96y)/0.8
x = 125.9 - 1.2y
Replace that in the other equation:
3*(125.9 - 1.2y) + 2y = 245
Solving for y:
3*125.9 - 3*1.2y + 2y = 245
y*(2 - 3*1.2) = 245 - 3*125.9
y = (245 - 3*125.9)/(2 - 3*1.2)
y = 82.94
Then the value of x is:
x = 125.9 - 1.2*82.94
x = 26.37
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Find the area of a triangle with the given base b and height h . b=6 in., h=15 in.
Answers
6 x 15 = 90
90/2 = 45
area = 45cm^2
6.) The volume of a sphere is two-thirds the volume of a cylinder with the same radius r and height of 2r.
True
False
Answers
express (2+4√6)÷(3√2-1) in the form p√2+q√3 where p and q are rational numbers
Answers
3^9/3^2
Simplify. Express using exponents
Please answer lol
Answers
Answer:
2187
Step-by-step explanation:
Answer:
3'7
Step-by-step explanation:
3'7x3'2=3'9
3'7x3'2/3'2=3'7
simplify 3'2/3'2
Use an algebraic approach to solve the problem.
Aura took three biology exams and has an average score of 87. Her second exam score was 11 points better than her first, and her third exam score was 5 points better than her second exam. What were her three exam scores?
Answers
Answer:
Her first exam score was a 78, her second exam score was a 89, and his third exam score was a 94.
Step-by-step explanation:
Use the formula for the mean: sum of elements / number of elements
Let x represent her first exam score.
Her second exam score can be represented by x + 11, since it was 11 points better than her first.
Her third exam score can be represented by (x + 11) + 5, since it was 5 points better than her second.
Plug in all of these expressions into the mean formula. Plug in 87 as the mean, and plug in 3 as the number of elements (since there are 3 scores):
mean = sum of elements / number of elements
87 = ( (x) + (x + 11) + (x + 11) + 5 ) / 3
Add like terms and solve for x:
87 = (3x + 27) / 3
261 = 3x + 27
234 = 3x
78 = x
So, her first score was a 78.
Find her second score by adding 11 to this:
78 + 11 = 89
Find her third score by adding 5 to the second score:
89 + 5 = 94
Her first exam score was a 78, her second exam score was a 89, and his third exam score was a 94.
The Oakland School's harvest festival is coming up, and Ms. Garza is in charge of ordering the caramel apples again. She knows she ordered 9 cases of caramel apples last year but can't remember how many apples come in a case. She checks last year's records and finds that the school sold 685 caramel apples and had 35 left over. Which equation can Ms. Garza use to find how many caramel apples, c, come in each case?
Answers
Answer:
9c-685=35
Step-by-step explanation:
9c is the number of total apples and 35 t 685 is the number of total apples
9c = 35t685
c=80
check: 9 x 80 - 685 = 35
Answer:
the answer is 80 carmel apples 9c-685=35
Step-by-step explanation:
weights of american adults are normally distributed with a mean of 180 pounds and a standard deviation of 8 pounds. what is the probability that a randomly selected individual will be between 185 and 190 pounds?proposed solution: pnorm(190-185,180,8)
Answers
There is 16.2% probability that a randomly selected individual will be between 185 and 190.
What is probability?
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are going to happen, using it. Probability can range from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.
What is Z score?
A z-score describes the position of a raw score in terms of its distance from the mean, when measured in standard deviation units. The z-score is positive if the value lies above the mean, and negative if it lies below the mean.
z = (x - μ) / σ
where μ is the mean, x = raw score and σ is the standard deviation.
Given,
mean = 180
standard deviation = 8 pounds.
For 185
z = (185 - 180)/8 = 0.625
for 190
z = (190 - 180)/8 = 1.25
P(185 < x < 190) = P(0.625 < z < 1.25) = P(z < 1.25) - P(z < 0.625) = 0.8944 - 0.7324 = 16.2%
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Express the formula d=rt in terms of the time,t. Use your formula to find the time when the distance is 40 and the rate is 8.
Answers
The expression for d=rt in terms of the time is t = d/r; t = 5.
What is time? Time can be defined as a continuous and ongoing sequence of events that occur consecutively from the past to the present to the future. Time is used to measure, measure or compare the duration of events or the intervals between them, and even the sequence of events. Time is a useful concept that we use in our daily life. We have to watch when we cook, play, study, go to school, meet someone, etc. So knowing the right time is very important. Time is usually the answer to when an event happens or happened. The concept of time determines when a certain event occurs, has occurred or will occur. Time is a measurable quantity and is also infinite. The time is calculated in seconds, minutes, hours, days, months and years.Therefore,
In the equation d=rt
t = d/r
when distance is 40 and the rate is 8
t = d/r
Replace d with 40 and rate with 8
t = 40/8
t = 5
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A parallelogram is cut out of a 12-inch by 8-inch sheet of paper. There are four right triangle remnants. Two have the dimensions 2 inches by 9 inches, and the other two have the dimensions 3 inches by 6 inches.
A parallelogram is shown. An altitude is drawn from one point to the opposite side to form a right angle. The length of the base is 9.22 inches.
The resulting parallelogram has a base of approximately 9.22 inches.
Complete the following steps to calculate the altitude of the parallelogram using area methods.
The area of the sheet of paper is
square inches.
The combined area of the triangle cutouts is
square inches.
The area of the parallelogram is
square inches.
The altitude of the parallelogram rounded to two decimals is
square inches.
Answers
The area of a shape is the amount of space it occupies.
The area of the paper is 96 square inchesThe combined area of the triangle cutouts is 36 square inchesThe area of the parallelogram is 60 square inchesThe altitude of the parallelogram is 6.51 inchesThe area of the paperThe dimension of the paper is given as: 12-inch by 8-inch
So, its area is:
\(Area = 12 * 8\)
\(Area = 96\)
Hence, the area of the paper is 96 square inches
The area of the trianglesThe dimensions of the 4 right triangles are: Two 2 inches by 9 inches, and two 3 inches by 6 inches
So, the combined area is:
\(Area = 0.5(2 * 2 * 9 + 2 * 3 * 6)\)
\(Area = 36\)
Hence, the combined area of the triangle cutouts is 36 square inches
The area of the parallelogramThe area of the parallelogram is the difference between the areas of the paper and the four right triangles.
So, we have:
\(Area = 96 - 36\)
\(Area = 60\)
Hence, the area of the parallelogram is 60 square inches
The altitude of the parallelogramThe base is given as 9.22
So, we have:
\(Area = Base * Altitude\)
This gives
\(60 = 9.22* Altitude\)
Divide both sides by 9.22
\(Altitude = 6.51\)
Hence, the altitude of the parallelogram is 6.51 inches
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Answer: The area of the paper is 96 square inches
The combined area of the triangle cutouts is 36 square inches
The area of the parallelogram is 60 square inches
The altitude of the parallelogram is 6.51 inches
Step-by-step explanation:
the way you speak, or your ___, Can let others know how you are feeling
A.tone
B.emotion
C.voice
D.language
Answers
Answer:
A, tone
Tone can also be used in poetry or writing to set the mood. It's kinda like your attitude, so this is the best answer.
Step-by-step explanation:
Answer:
Tone
Step-by-step explanation:
Tone matters more than word choice.
this is my third time asking brainly for this question. Find the perimeter and in standard form.
Answers
according to researchers, what is the percentage of high school seniors and eighth grade students, respectively, who have used marijuana in the past year?
Answers
The percentage of high school seniors and eighth grade students is the 40% and the 15%.
According to the statement
we have to find that the percentage of high school seniors and eighth grade students.
For this purpose, we know that the
The percentage is a ratio expressed as a fraction of 100.
And with the given information,
the percentage is the used by the marijuana in the past year.
By the researchers,
the percentage of high school seniors in 2011 is 40%
And
the percentage of eighth grade students is 15%.
So, The percentage of high school seniors and eighth grade students is the 40% and the 15%.
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please it do in 6 min i need help with this one please The following points are from a dilation. If
B(0 , 5 ) and B'(x, 2 0 ), what is x?
Answers
Answer:
The value of x = 0
Step-by-step explanation:
Given
B(0, 5)B'(x, 20)To determine
The value of x = ?
If we closely observe the value of the y-coordinate of the image B'(x, 20), we can see that the y-coordinate of the image B' is 4 times the y-coordinate of the original point (0, 5).
i.e. y → 5×4 =20
It means the coordinates of the image B'(x, 20) can be determined by dilating the original point (0, 5) by a scale factor of 4.
Thus, the x-coordinate of B'(x, 20) can also be obtained by multiplying the x-coordinate of B(0, 5) by 4.
i.e. x → 4×0 = 0
Thus, the rule of dilation is:
P(x, y) = P'(4x, 4y)
Hence,
B(0, 5) → B'(4(0), 4(5)) → B'(0, 20)
Thus,
The value of x = 0 ∵ 0×4 = 0
An amusement park offers a ticket for $12.95 off the original price p. Write and solve an equation to find the original price
Answers
The original price of a ticket is a $56.95.
According to the statement
We have to find that the linear equation.
So, For this purpose, we know that the
The definition of a linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.
From the given information:
An amusement park offers a ticket for $12.95 off the original price p.
Then
Discounted price is that the amount of sum that extracted from the initial price when buyers receive some style of leverage from the vendor. during this case, discounted price of the ticket is $44 which was $12.95 but the initial price.
Then the first price would be:
$44 + $12.95 = $56.95
So, The original price of a ticket is a $56.95.
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