The correct minimum number of students required to guarantee that at least three answer sheets must be identical is 5.
To solve this problem, we can use the pigeonhole principle. The pigeonhole principle states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon.
In this scenario, each question can be thought of as a pigeonhole, with the possible answers (a, b, c, d) as the pigeons. Since there are four possible answers for each question, we have four pigeonholes for each of the ten questions.
Now, let's consider the number of students (pigeons) and the number of unique answer sheets (pigeonholes). We want to find the minimum number of students such that at least three of them have identical answer sheets.
For two students, there are [tex]\(4^{10}\)[/tex] possible combinations of answers for the ten questions. For three students, the number of possible combinations of answers is [tex]\(4^{10} \times 4^{10}\)[/tex]. In general, for [tex]\(n\)[/tex] students, the number of possible combinations is [tex]\(4^{10} \times 4^{10} \times \ldots \times 4^{10}\) (\(n\) times)[/tex].
We need to find the smallest [tex]\(n\)[/tex] such that the number of combinations exceeds the number of ways to distribute the answer sheets without having three identical ones.
The number of ways to distribute the answer sheets without having three identical is given by the sum of the combinations of choosing 1, 2, ...,[tex]\(n-1\)[/tex] students out of [tex]\(n\)[/tex] to have unique answer sheets, and then multiplying by [tex]\(4^{10}\)[/tex] for each unique combination. This is because we are allowing for the possibility that some students have the same answer sheets, but not more than two.
The sum is given by:
[tex]\[ \binom{n}{1} \times 4^{10} + \binom{n}{2} \times 4^{10} + \ldots + \binom{n}{n-1} \times 4^{10} \][/tex]
We want to find the smallest [tex]\(n\)[/tex] such that:
[tex]\[ \binom{n}{1} \times 4^{10} + \binom{n}{2} \times 4^{10} + \ldots + \binom{n}{n-1} \times 4^{10} < 4^{10n} \][/tex]
For [tex]\(n = 4\)[/tex], we have:
[tex]\[ \binom{4}{1} \times 4^{10} + \binom{4}{2} \times 4^{10} + \binom{4}{3} \times 4^{10} = 4 \times 4^{10} + 6 \times 4^{10} + 4 \times 4^{10} = 14 \times 4^{10} \][/tex]
For [tex]\(n = 5\)[/tex], we have:
[tex]\[ \binom{5}{1} \times 4^{10} + \binom{5}{2} \times 4^{10} + \binom{5}{3} \times 4^{10} + \binom{5}{4} \times 4^{10} \][/tex]
[tex]\[ = 5 \times 4^{10} + 10 \times 4^{10} + 10 \times 4^{10} + 5 \times 4^{10} = 30 \times 4^{10} \][/tex]
Now, we compare [tex]\(30 \times 4^{10}\) with \(4^{10 \times 5}\)[/tex]. Since [tex]\(4^{10 \times 5}\)[/tex] is much larger than[tex]\(30 \times 4^{10}\)[/tex], we can see that for [tex]\(n = 5\)[/tex], the number of possible combinations exceeds the number of ways to distribute the answer sheets without having three identical ones.
Therefore, the minimum number of students required to guarantee that at least three answer sheets must be identical is 5.
Paisley needs 32 in of ribbon for a hair bow. The ribbon is sold by centimeters. Givin that 1 in = 2.54 centimeters, how many centimeters of ribbon does she need.
Work Pls
Thx
5-2i and 7+6i find the midpoint
The midpoint between the complex numbers 5-2i and 7+6i is 6 + 2i.
Explanation:To find the midpoint between two complex numbers, we can use the formula:
Midpoint = (Re(z1) + Re(z2))/2 + (Im(z1) + Im(z2))/2i
Applying the formula to the given complex numbers 5-2i and 7+6i, we have:
Midpoint = (5 + 7)/2 + (-2 + 6)/2i
Midpoint = 12/2 + 4/2i
Midpoint = 6 + 2i
https://brainly.com/question/11839107
#SPJ3
ΔABC and ΔXYZ are similar triangles. If BC = x + 7, AC = x + 6, YZ = 4 − x, and XZ = 3 − x, find the value of x.
Answer:
-3/2 or -1.5
Step-by-step explanation:
-3/2 = -1.5
BC = -1.5 + 7 = 5.5
AC = -1.5 + 6 = 4.5
YZ = 4 - (-1.5) = 5.5
XZ = 3 - (-1.5) = 4.5
what is the equation for horizontal line through the point (0,3)?
Which of the following is the prime factored form of the lowest common denominator of?
A.2 x 2 x 2 x 3
B.2 x 4 x 2 x 3
C.8 x 6
D.2 x 12
2 x 2 x 2 x 3 factored form is the prime factored form.
What is a prime number?"A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers."
The given factored forms are:
A. 2 x 2 x 2 x 3
Here, all the numbers are prime numbers.
Therefore, this factored form is the prime factored form.
B. 2 x 4 x 2 x 3
Here, all the numbers are not prime numbers.
Therefore, this factored form is not the prime factored form.
C. 8 x 6
Here, not a single number is prime number.
Therefore, this factored form is not the prime factored form.
D. 2 x 12
Here, all the numbers are not prime numbers.
Therefore, this factored form is not the prime factored form.
Learn more about prime number here: https://brainly.com/question/10454590
#SPJ3
The equilateral triangle shown is rotated about line a. Each side of the triangle measures 20 mm. What shape is created by the rotation and what is the approximate circumference of the base? Circumference of a circle: C = 2πr a cylinder with a circumference of about 63 mm a cylinder with a circumference of about 126 mm a cone with a base circumference of about 63 mm a cone with a base circumference of about 126 mm
The shape and the circumference of the base created by the rotation is; Cone and Circumference of 63 mm
What shape is created by rotation?We are given;
Each side of equilateral triangle = 20 mm
Now, when the triangle is rotated about the line ‘a’ which passes through the midpoint of any of the 3 sides of the equilateral triangle we will get a conical shape.
The distance from the line a which cuts one side of a triangle to one vertex of the triangle is the radius = 10 mm
We know that;
Circumference of a circle = 2πr
Thus;
C = 2 × 3.14 × 10
C = 6.28 × 10
C = 62.8 mm
C = 63 mm
Thus, the approximate circumference of the base is 63 mm.
Read more about rotation at; https://brainly.com/question/4289712
the cost of internet access at a cafe is a function of time the cost for 8, 25, and 40 minutes are shown write the equation in slope-intercept form that represents the function then find the cost of surfing the web at the cafe for one hour
Answer:
?
Step-by-step explanation:
Where is the graph?
Can somebody solve this system equation using addition please?
-4x-5y=7
3x+5y=-14
I can’t solve it and it’s due tomorrow for my math class also please show work on how you’ve solved it
-4x - 5y = 7
3x + 5y = -14
You can add these two equations together straightaway since the y-terms have opposite coefficients.
-4x - 5y = 7
3x + 5y = -14
+___________
-x - 0 = -7
-x = -7
x = 7
Substitute 7 for x into either of the original equations and solve algebraically to find y.
3x + 5y = -14
3(7) + 5y = -14
21 + 5y = -14
21 = -14 - 5y
35 = -5y
-7 = y
Finally, check work by substituting both x- and y-values into both original equations.
-4x - 5y = 7
-4(7) - 5(-7) = 7
-28 + 35 = 7
7 = 7
3x + 5y = -14
3(7) + 5(-7) = -14
21 - 35 = -14
-14 = -14
Answer:
x = 7 and y = -7; (7, -7).
help me pls i need to pass
The surface area (S) of a rectangular prism is given in terms of its length (L), width (W), and height (H) as
... S = 2(LW +WH + LH)
Substitute your numbers and do the arithmetic.
... S = 2(7.2·10 + 10·6.3 + 7.2·6.3)
... = 2(72 + 63 + 45.36)
... = 2·180.36
... = 360.72 . . . . square inches
Suppose that the demand equation for a certain item is 1p+1x2−160=0. evaluate the elasticity at 65:
The required value of elasticity at x = 65 is -0.007.
Given that,
The demand equation for a certain item is [tex]1p + 1x^{2} -160 = 0[/tex].
We have to determine,
The value of elasticity at 65.
According to the question,
The price elasticity of demand is calculated as the percentage change in quantity divided by the percentage change in price.
Therefore,
Price of elasticity = [tex]\frac{dp}{dx}[/tex]
Calculate change by differentiation with respect to p:
[tex]\dfrac{d(p+x^{2} -160)}{dx}\\\\1 + 2x\frac{dp}{dx} = 0\\[/tex]
Then,
[tex]E(x) = \frac{dp}{dx} = \frac{-1}{2x}[/tex]
Therefore, The elasticity at x = 65 is ,
[tex]E(65) = \frac{-1}{2(65)} = -0.007[/tex]
Hence, The required value of elasticity at x = 65 is -0.007.
For more information about Elasticity click the link given below.
https://brainly.com/question/14547451
the circumference of an Oreo is 6.28 inches what is the area of the top cookie?
The circumference of an Oreo is 6.28 inches what is the area of the top cookie?
Solution:
Circumference= 6.28 inches
The formula of circle =2πr
So, 2πr=6.28
2*3.14*r=6.28
6.28r=6.28
To find the value of r, Let us divide by 6.28 on both sides
6.28 r /6.28 =6.28/6.28
r=1
Area of circle=πr²
=π1²
=π
=3.14 inches²
Area of the top cookie=3.14 inches²
John likes 400 but not 300; he likes 100 but not 99; he likes 3600 but not 3700. Which does he like?
o 900
o 1000
o 1100
o 1200
Vicente is watching a movie that last 1 hour and 37 minutes you watch 52 minutes of our how many minutes are left in the movie
a band that will be the opening act for a concert charges a $700.00 flat fee as well as 16% of all ticket profits from ticket sales. if the band earned $1,200, how much was made in total ticket sales for the show?
A marketing firm conducts a survey to find out how many people use a product. One hundred people are contacted for the study and fifteen confirm that they use the product. Which describes the survey?
The marketing firm is going forward in time and observing groups sharing common factors.
The marketing firm is applying some treatment to the subjects and then proceeding to observe its effect on the subjects.
The marketing firm is observing and measuring specific characteristics of the subjects, but is not attempting to modify the subjects.
The marketing firm is going back in time to collect data over some past period
Final answer:
The marketing firm's survey, where a sample of people is asked if they use a product, is an example of survey research aimed at observing and measuring characteristics without attempting modification. This cross-sectional study helps understand consumer behavior. (Option C)
Explanation:
The survey described by the marketing firm where one hundred people are contacted and fifteen confirm that they use the product is an example of survey research, which is a quantitative research method. In this survey, the firm is observing and measuring specific characteristics of the subjects, but not attempting to modify the subjects. This type of survey can help the firm in understanding consumers' preferences, tastes, attitudes, and behaviors regarding the use of the product. They are not applying any treatment to the subjects or observing them over time as would be done in a longitudinal study. Instead, they're conducting a cross-sectional survey at a single point in time. (Option C)
A test has forty questions worth 100 points.The test consists of True/False questions worth 3 points each and multiple choice questions worth 2 points each. How many multiple choice questions are on the test?
Answer:
20 multiple choice questions
Step-by-step explanation:
I hope this help you out
Last year 40 people adopted a manatee through Fia's Foundation.This year 30% more people adopted a manatee. How many more people adopted a manatee this year ?
A package of three compact discs cost $24.96. How much does each disc cost?
What is the value of y?
11.6 ft
12.5 ft
16.5 ft
17.5 ft
If a bank has total deposits of $7,700 and reserves of $2,100: instructions: round to the nearest whole number. (a) what is its current reserve ratio? % (b) how large is its loan portfolio (outstanding loans)? %
A cyclist rides a bicycle with a wheel radius of 0.500 m across campus. a piece of plastic on the front rim makes a clicking sound every time it passes through the fork. if the cyclist counts 320 clicks between her apartment and the cafeteria, how far has she traveled? (a) 0.50 km (b) 1.5 km (c) 1.0 km (d) 0.80 km (e) 1.8 km
What is the area of the figure?
given that ABCD= PQRS, find AD
A.3
B.5
C.6
D.10
The probabilities for four different simple events are given. Of the four, which is MOST LIKELY to occur?
A. 0.5
B. 0.9
C. 0.35
D. 0.0100
The current size of an image on Samuel's computer is shown below: A rectangle is shown. The length of the rectangle is labeled as length equal to 8 inches and the width is labeled as width equal to 10 inches. What are the dimensions of the image at fraction 1 over 2 times its current size
Answer:
The length and the width of image is 4 inches and 5 inches respectively
Step-by-step explanation:
Given : Length of rectangle is 8 inches
Width of rectangle is 10 inches
To Find: What are the dimensions of the image at fraction [tex]\frac{1}{2}[/tex] times its current size
Solution:
Length of rectangle is 8 inches
Since we are given that the dimensions of the image at fraction [tex]\frac{1}{2}[/tex] times its current size
So, Length of image = [tex]\frac{1}{2} \times 8 = 4[/tex]
Thus the length of image is 4 inches.
Width of rectangle is 10 inches
Since we are given that the dimensions of the image at fraction [tex]\frac{1}{2}[/tex] times its current size
So, width of image = [tex]\frac{1}{2} \times 10 = 5[/tex]
Thus the width of image is 5 inches.
Hence the length and the width of image is 4 inches and 5 inches respectively.
3^2·3^5·3^-3 Please help.
Calculate the taxes based on the following taxable income amounts
I really need help on this fast please!
I need the estimated taxes for each problem at the bottom is the table needed.
1) income of 90,714.00
2) income of 63,036.00
3) income of 220,596.00
4) income of 166,904.00
5) income of 220,770.00
6) income of 97,266.00
7) income of 207,264.00
8) income of 479,956.00
9) income of 511,195.00
10) income of 511,104.00
11) income of 212,817.00
12) income of 435,045.00
13) income of 655,952.00
14) income of 739,334.00
15) income of 322,335.00
Income range Taxes
From To Tax rate. Plus
0.00 8,925.00 10.0%.
8925.00 36,250.00 15.0% 892.50
36,250.00 87,850.00 25.0% 5,437.50
87,850.00 183,250.00 28.0% 21,962.50
183,250.00 398,350.00 33.0% 51,310.00
398,350.00 400,000.00 35.0%. 131,455.50
400,000.000 - 39.6%. 140.000.00
Q10 Q2.) Find the standard form of the equation of the parabola satisfying the given conditions.
which constant term would mean that the expression is completely factored x ^ 2 - 3x +
-10
0
10
To completely factorize [tex]\(x^2 - 3x + c\),[/tex]the correct constant term is 10. With this, the expression becomes [tex]\((x - 2)(x - 5)\),[/tex] achieving complete factorization.
To find the correct constant term that would allow for complete factorization of the quadratic expression [tex]\(x^2 - 3x + c\),[/tex] let's consider what it means to factorize a quadratic expression:
A quadratic expression can be factored if it can be represented in the form [tex]\((x - a)(x - b)\),[/tex] where a and b are the roots of the expression. Given the original expression [tex]\(x^2 - 3x + c\),[/tex] we can expand [tex]\((x - a)(x - b)\)[/tex] and then compare coefficients to determine the constant term c.
Expanding [tex]\((x - a)(x - b)\)[/tex] :
- [tex]\((x - a)(x - b) = x^2 - (a + b)x + a \cdot b\).[/tex]
Comparing Coefficients :
- By comparing with [tex]\(x^2 - 3x + c\),[/tex] we can identify that [tex]\(a + b = 3\)[/tex] (the coefficient for [tex]\(x\)[/tex] and [tex]\(a \cdot b = c\)[/tex] (the constant term).
- Given \(a + b = 3\), let's find possible values for a and b that would yield a correct factorization:
- Consider [tex]\(a = 1\), \(b = 2\):[/tex] Then [tex]\(a \cdot b = 1 \times 2 = 2\),[/tex] which is different from the given constant \(c\).
- Consider [tex]\(a = -2\), \(b = -5\):[/tex] Then [tex]\(a \cdot b = -2 \times -5 = 10\),[/tex] suggesting that [tex]\(c = 10.[/tex]
- Consider [tex]\(a = 5\), \(b = 2\):[/tex] Then [tex]\(a \cdot b = 5 \times 2 = 10\),[/tex] also suggesting [tex]\(c = 10\).[/tex]
Considering this process, the correct answer that would allow for complete factorization of the given expression [tex]\(x^2 - 3x + c\)[/tex] is 10:
Thus, the factorization of [tex]\(x^2 - 3x + 10\)[/tex] results in [tex]\((x - 2)(x - 5)\),[/tex] suggesting that the constant term in question should be 10.
The complete question is : Which constant term in the expression [tex]\(x^2 - 3x + c\)[/tex] would allow it to be completely factored? Consider the possible values of c and determine which would result in a fully factored form. Options: -10, 0, 10.
The correct constant term that completes the factoring of the expression [tex]\( x^2 - 3x + \ ? \)[/tex] is 10.
To find the constant term that completes the factoring of the quadratic expression [tex]\( x^2 - 3x + \ ? \)[/tex], we can follow these steps:
Understand that when factoring a quadratic expression of the form [tex]\( ax^2 + bx + c \)[/tex], we are looking for two numbers that multiply to ( ac ) and add to ( b ).
In our case, ( a = 1 ), ( b = -3 ), and ( c ) is the constant term we're looking for.
Since ( a ) is 1, we need to find two numbers that multiply to ( c ) and add up to ( -3 ).
These two numbers are the factors of ( c ).
Given that the constant term ( c ) is the term that doesn't include ( x ), it will be the product of these two factors.
To find the constant term, we can factorize the expression ( ac ), where ( a = 1 ) and ( c ) is the constant term.
Once we find the factors of ( c ), we can test different values until we find the correct one that makes the expression factorable.
Let's start by factoring ( ac ):
Since ( a = 1 ), and ( b = -3 ), we have ( ac = c ).
Given that the product of the factors should be ( c ), and the factors should add up to ( -3 ), we can find the factors of ( c ) by trial and error.
Let's try different values of ( c ) and see which one works:
If ( c = 10 ), the factors of ( c ) would be 1 and 10. However, 1 + 10 = 11, not -3.
If ( c = -10 ), the factors of ( c ) would be -1 and 10. However, -1 + 10 = 9, not -3.
If ( c = -10 ), the factors of ( c ) would be 1 and -10. However, 1 + (-10) = -9, not -3.
If ( c = 10 ), the factors of ( c ) would be -1 and -10. However, -1 + (-10) = -11, not -3.
If ( c = 0 ), the factors of ( c ) would be 0 and 0. However, 0 + 0 = 0, not -3.
Based on these trials, we see that none of the values satisfy the condition of adding up to -3.
Therefore, the correct constant term that completes the factoring of the expression [tex]\( x^2 - 3x + \ ? \)[/tex] is 10.
You own a coffee shop where a cup of coffee costs $2.10. Your cost on the cup of coffee is $0.30. Calculate the margin per cup of coffee.