) there are exactly 20 students currently enrolled in a class. how many different ways are there to pair up the 20 students, so that each student is paired with one other student

Answers

Answer 1

Assuming that "pair up students" means "divide up all 20 of the students into groups of two," and we regard two pairings as the same if and only if, in each pairing, each student has the same buddy, then I believe that your answer of 20! / [(2!)^10 * (10!)] is correct. (And I also believe that this is the best interpretation of the problem as you've stated it.)

There are at least two ways to see this (possibly more).

One way is to note that, first, we have to select 2 students for the first pair; that's C(20, 2) (where by C(20, 2) I mean "20 choose 2"; that is, 20! / (18! * 2!). )

Then, for each way of selecting 2 students for the first pair, I have to select 2 of the remaining 18 students for the second pair, so I multiply by C(18, 2).

Continuing in this manner, I get C(20, 2) * C(18, 2) * ... * C(2, 2).

But it doesn't matter in this situation the order in which I pick the pairs of students. Since there are 10! different orders in which I could pick the individual pairs, then I want to divide the above by 10!, giving me the answer

[C(20, 2) * C(18, 2) * ... * C(2, 2)] / 10!.

This is the same as your answer, because C(n , 2) = n(n - 1) / 2, so we can simplify the above as

[(20 * 19) / 2 * (18 * 17) / 2 * ... * (2 * 1) / 2] / 10!
= 20! / [2^10 * 10!]
= 20! / [(2!)^10 * (10!)].

Another way is to reason as follows:

1. First, arrange the 20 students in a line; there are 20! ways to do this
2. We can get a pairing by pairing the 1st and 2nd students in line together, the 3rd and 4th students together, etc.
3. But if I switch the order of the 1st and 2nd student, then this doesn't give a different pairing. I don't want to count both orderings separately, so I divide by 2!
4. The same argument from step 3 holds for the 3rd and 4th student, the 5th and 6th student, etc., so I need to divide by 2! nine more times
5. Finally, the particular order in which I selected the ten pairings are unimportant--for example, the following orderings don't produce different pairings:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
3, 4, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

So I need to further divide by the number of ways I can arrange the ten pairs, which is 10!.

Related Questions

Please, show me how to solve this. Find the limit as x approaches −8 for the function f(x)=5x+12.

Answers

ahemm... cheap answer is just [tex]\bf \lim\limits_{x\to -8}~5x+12\implies 5(-8)+12\implies -28[/tex]

Suppose that 19 inches of wire costs 95 cents. At the same rate, how much (in cents) will 37 inches of wire cost?

Answers

0.95/19 = 0.05 cents per inch

37*0.05 = 1.85

it will cost $1.85

Will someone please answer this??

Answers

7.2 Feet Should be the correct answer

Five individuals, including a and b, take seats around a circular table in a completely random fashion. suppose the seats are numbered 1, . . . , 5. let x = a's seat number and y = b's seat number. if a sends a written message around the table to b in the direction in which they are closest, how many individuals (including a and
b.would you expect to handle the message?

Answers

Will use A and B in place of a and b for clarity.
Let x=number of individuals away from A, including A & B

Without loss of generality, assume A is seated in seat #1.

Then B is seated at 2,3,4,5 with equal probability.
Half of the time B is seated at 2 or 5, each of which is next to A, therefore x=2
The other half of the time B is seated at 3 or 4, each of which is separated from A by one seat, then x=3.

The expected number of individuals
E[X]=sum (x*P(x))
=2*(1/2)+3(1/2)
=2.5

So the expected number of individuals to handle the message is 2.5.

The number of individuals you would expect to handle the message is 2.5.

Joint probability distribution

Let Z represent the number of individuals that handle the message

Table for the possible joint value of X and Y

Z Y

1 2 3 4 5

X 1 - 2 3 3 2

2 2 - 2 3 3

3 3 2 - 2 3

4 3 3 2 - 2

5 2 3 3 2 -

Each cell contain=1/4×1/5=1/20

Hence:

Number of individual=10×2×1/20+10×3×1/20

Number of individual=20×0.05+30×0.05

Number of individual=2.5

Therefore the number of individuals you would expect to handle the message is 2.5.

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Your science quiz had 17 questions and you answered 13 of the questions correctly. What is your present score?

Answers

dived 13 by 17

13/17 = 0.7647

= 76.47%

if you need to round the number it would be 76%, which is a 76 grade

factoring
x^2-4x-21=0

Answers

Find two numbers that add to -4 and that multiply to -21.

After playing around with some numbers, I got -7 and 3.

So, the factored form is (x - 7)(x + 3).

A local carpet company has been hired to carpet a planetarium which is in the shape of a circle. If the radius of the planetarium is six yards, and the cost of the carpet is $14 per square yard, find the total cost to carpet the planetarium.

Answers

The cost of the carpet will be given by:
cost=[area of the carpet]*[price per yard]
area of the carpet will be given by:
area=πr^2
=π*6^2
=113.1 square yards
thus the cost of the carpet will be:
113.1*14
=$1,583.4

A potter use 4/5 of a pound of clay to make a bowl.How many bowls can the potter make from 12 pounds.

Answers

12/1 / 4/5=

12/1 * 5/4 =

60/4 = 15

they can make 15 bowls

Triangle XYZ is reflected across the line x = 3. What is the reflection image of X?

Answers

The image of x is the point (7,5).
Draw the line x= 3 and treat it like a mirror.

When you reflect the points of the triangle just count how many units each point is from the mirror (x=3)
Count the same number of points in the opposite direction (away from the mirror) and you will arrive at the coordinates for the points

Answer:

It's (7,5)

Step-by-step explanation:

Which number is a prime number? 21,22,23,24

Answers

23 is the prime number bc it isn't divisable by anything

A prime number is a number that has no factors, except for itself and one.

21 is not a prime number, because in addition to having one and itself as factors, it has 7 and 3. ( 1 x 21 = 21.) ( 7 x 3 = 21.)

22 is not a prime number. It's easy to tell because it is an even number. ( 1 x 22 = 22.) ( 2 x 11 = 22.)

24 is not a prime number. It's even, but also has more factors. ( 2 x 12 = 24.) ( 3 x 8 = 24.) (4 x 6 = 24.)

23, however, has no other factors but one and itself. ( 1 x 23 = 23.) So therefore, it is prime.

Write as a single power: 4^20 + 4^20 + 4^20 + 4^20

Answers

Sorry, I misinterpreted the question before.\\\\ 4^20+4^20+4^20+4^20 \\\\ 4(4^20)\\\\ 4^21\\\\

Owners of a recreation area are filling a small pond with water. They are adding water at a rate of 35 liters per minute. There are 700 liters in the pond to start. Let W represent the total amount of water in the pond (in liters), and let T represent the total number of minutes that water has been added. Write an equation relating W to T . Then use this equation to find the total amount of water after 19 minutes.

Answers

W=700+35T. Then replace T with 19 which will be W=700+35(19) which equals to 1365 total amount of water in liters.

An equation relating W to T is,

W = 700 + 35T

And, Total amount of water after 19 minutes is, 1365 liters

We have to given that,

Owners of a recreation area are filling a small pond with water. They are adding water at a rate of 35 liters per minute.

There are 700 liters in the pond to start.

Now, Let W represent the total amount of water in the pond (in liters), and let T represent the total number of minutes that water has been added.

Hence, an equation relating W to T is,

W = 700 + 35T

So, For T = 19 minutes,

W = 700 + 35 x 19

W = 700 + 665

W = 1365

Therefore, Total amount of water after 19 minutes is, 1365 liters

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Write the equation in spherical coordinates. 3x + 2y + 3z = 1

Answers

[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\cos\varphi\end{cases}[/tex]

[tex]3x+2y+3z=1[/tex]
[tex]\implies3\rho\cos\theta\sin\varphi+2\rho\sin\theta\sin\varphi+3\rho\cos\varphi=1[/tex]
[tex]\implies\rho=\dfrac1{(3\cos\theta+2\sin\theta)\sin\varphi+3\cos\varphi}[/tex]

A jeep and BMW enter a highway running east-west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.

Answers

recall you d = rt, distance = rate * time

when the BMW has been running for 2hrs, the Jeep has been running for 2.5hrs, because it started off the exit 30minutes(0.5hr) before the BMW.

if the rate of the BMW is say "r", then the rate of the Jeep is "r - 7", slower by 7mph.

now, we know after the Jeep was running for 2.5hrs and the BMW was running for 2hrs, they were both apart by 306.5miles, so, say if the Jeep travelled a distance of "d", then the BMW travelled the slack from the 306.5, or "306.5 - d".

[tex]\bf \begin{array}{lccclll} &distance&rate& \begin{array}{cllll} time\\ (hours)\\ \end{array}\\ &-----&-----&-----\\ Jeep&d&r-7&2.5\\ BMW&306.5-d&r&2 \end{array} \\\\\\ \begin{cases} \boxed{d}=(r-7)2.5\\ 306.5-d=2r\\ ----------\\ 306.5-\boxed{(r-7)2.5}=2r \end{cases}[/tex]

solve for "r" to get the rate of the BMW.

what about the Jeep's? well, the Jeep is just r - 7.

Answer:

3x+27+3.5x=397.5

6.5x=370.5

x=57 mph for jeep and travels 199.5 miles in 3.5 hours

x+9=66 mph for B and travels 198 miles in 3 hours

Step-by-step explanation:

speed of jeep=x, time is >>>>> t+0.5 in hours or 3.5 hours here

speed of B=x+9, time is >>>>>>>>>>>> 3 hours

distance is >>>>>>>>> speed time

Hope it Helps.

Answer on: june 11, 2021

The scores on an exam are normally distributed, with a mean of 74 and a standard deviation of 7. What percent of the scores are less than 81?

Answers

Mean = 74
Standard deviation = 7

For 81%, the Z-score is
Z=(X-mean)/(standard deviation)
=(81-74)/7
=1

So look up table of normal distribution for
P(Z<1)=0.8413
=>
On average, 84% of scores are less than 81.

The marching band is selling cases of fruit for $13 per case. (a) Write an algebraic expression for the cost of f cases of fruit. (b) Evaluate the expression for 250 cases.

Answers

(a) Each case of fruit costs $13
∴ $13 × f cases
= 13f

(b) Since f = 250
= 13(250)
= $3250

An algebraic expression for the cost of f cases of fruit will be (a) z = 13f and the cost for 250 cases will be (b) $3250.

How to form an equation?

Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.

In other words, an equation is a set of variables that are constrained through a situation or case.

Let's say the cost of the fruit is z

Given,

The marching band is selling cases of fruit for $13 per case.

So for f cases of fruit

z = 13f

And cost of 250 cases = z = 13(150) = $3250

Hence, the algebraic expression for the cost of f cases of fruit will be z = 13f and the cost for 250 cases will be $3250.

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Making use of the fact hat eq. (6.20) is an exact differential expression, show that what is the result of application of this equation to an ideal gas

Answers

Since there is no figure attached, I will describe the derivation of the ideal gas law. The combined gas law has no official founder; it is simply the incorporation of the three laws that was discovered. The combined gas law is a gas law that combines Gay-Lussac’s Law, Boyle’s Law and Charle’s Law. Boyle’s law states that pressure is inversely proportional with volume at constant temperature. Charle’s law states that volume is directly proportional with temperature at constant pressure. And Gay-Lussac’s law shows that pressure is directly proportional with temperature at constant volume. The combination of these laws known now as combined gas law gives the ratio between the product of pressure-volume and the temperature of the system is constant. Which gives PV/T=k(constant). When comparing a substance under different conditions, the combined gas law becomes P1V1/T1 = P2V2/T2.

What is the answer to this question?

Answers

9/12= 0.75

8.00 * 0.75 = 6.00

the 9" costs $6.00

If marc ABC = 184°, what is m∠ABC?

Answers

the angle is 1/2 of the arc

184/2 = 92 degrees

It would be half of the intercepted arc which is

360-184 = 176

176/2 = 88 degrees

Zooey predicts the movie will be 90 minutes long. If the movie actually is 102 minutes long, what is Zooey's percent error? Round your answer to the nearest tenth of a percent.

Answers

102 - 90 = 12
12 / 102 = 0.1176 = 11.76 rounds to 11.8% <==

%error=100(prediction-actual)/actual

%error=100(90-102)/102

%error≈ -11.8%

Now the negative sign indicates that she underestimated the length of he movie by 11.8%, but technically the percent error is an absolute value so it is just 11.8% error.

Is this statemate true or false?
All parallelograms are special kinds of squares.

Answers

This is false.
By definition, all squares must have four right angles. Not all parallelograms meet this requirement,

which is larger 2/3" x 3-7/16" or 0.6"L x 3.43"W?

Answers

2/3 x (3 - 7/16) = (2/3) * (3+7/16)
= (2/3)*(55/16)
= 55/24
= 2.2917

0.6 x 3.43
= 2.058

The first answer is greater than the second.

Answer: 2/3 x 3-7/16 is larger.

Help.. :)

Which equation is not equivalent to the formula e = mc?
m equals e over c
c equals e over m
e = cm
m equals c over e
Please help THANKS!

Answers

m equals c over e is not equal to e=mc

Answer with Step-by-step explanation:

we are given a equation:

e=mc

We have to find which equation is not equivalent to the above formula.

e=mc

Dividing both sides by c,we get

m=e/c

i.e. m equals e over c

e=mc

Dividing both sides by m,we get

c=e/m

i.e. c equals e over m

e=mc=cmBut m is not equal to c over e

Hence, The equation which is not equivalent to e=mc is:

m equals c over e

A carpenter is framing a window with wood trim where the length of the window is 9 1/3 feet. If the width of the window is 6 3/4 feet, how many feet of the wood is needed to frame the window?

Answers

This is not much of a question, because it doesn't take into account the width of the wood trim.

So you are asked to find the perimeter of the window, ignoring the width of the trim.

P = Perimeter = 2L + 2W.

Here, P= 2(9 1/3 feet) + 2(6 3/4 feet) = 2(28/3 feet) + 2(27/4 feet)

So P = 2[28/3 + 27/4] feet. LCD is 3*4 = 12.

Thus, P = 2 [ 28/3 + 27/4 ] feet. Can you finish? Add together the fractions 28/3 and 27/4.

what is the answer ?

Answers

The system of inequalities is the following:

i) y ≤ –0.75x
ii)y ≤ 3x – 2

since [tex]0.75= \frac{75}{100}= \frac{3}{4} [/tex], we can write the system again as

[tex]i) y \leq - \frac{3}{4}x [/tex]
[tex]ii) y \leq 3x-2[/tex]

Whenever we are asked to sketch the solution of a system of linear inequalities, we:

1. Draw the lines
2. Color the regions of the inequalities.
3. The solution is the region colored twice.

A.

to draw the line [tex] y =- \frac{3}{4}x[/tex]

consider the points: (-4, 1) and (0, 0), or any 2 other points (x,y) for which [tex] y =- \frac{3}{4}x[/tex] hold.

since we have an "smaller or equal to" inequality, the line is a solid line (not dashed, or dotted).

In order to find out which region of the line to color, consider a point not on the line, for example P(1, 1), which is clearly in the upper region of the line.

For (x, y)=(1, 1) the inequality [tex]y \leq - \frac{3}{4}x[/tex], does not hold because

[tex]1 \leq - \frac{3}{4}*1= -\frac{3}{4} [/tex] is not true,

this means that the solution is the region of the line not containing (1, 1), as shown in picture 1.

B.
similarly, to draw the solution of inequality ii) y ≤ 3x – 2,

we first draw the line y=3x-2, using the points (0, -2) and (2, 4), or any other 2 points (x,y) for which y=3x-2 holds.

after we draw the line, we can check the point P(1, 7) which clearly is above the line y=3x-2.

for (x, y) = (1, 7), the inequality y ≤ 3x – 2 does not hold

because 7 is not ≤ 3*1-2=1, so the region we color is the one not containing P(1, 7), as shown in picture 2.

The solution of the system is the region colored with both colors, the solid lines included. Check picture 3.

the lines intersect at (0.533, -0.4) because:

–0.75x=3x-2
-0.75x-3x=-2
-3.75x=-2, that is x= -2/(-3.75)=0.533

for x=0.533, y=3x-2=3(0.533)-2=-0.4

Answer: Picture 3, the half-lines included. So the graph is in the 3rd and 4th Quadrants

If f(x) is a nth degree polynomial then F^(n+1)(x)=0. True or false and why

Answers

Final Answer:

The given statement “If f(x) is a nth degree polynomial then [tex]F^{(n+1)(x)=0[/tex]. True or false and why” is false because the (n+1)st derivative being zero is contingent on the specific value of the leading coefficient in the polynomial. It is not a general rule for all nth degree polynomials.

Explanation:

The statement [tex]$F^{{(n+1)}(x) = 0$[/tex] is not universally true for all nth degree polynomials (f(x)). To understand why, consider a general nth degree polynomial [tex]$f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$[/tex], where [tex]$a_n$[/tex] is the leading coefficient and [tex]$a_n \neq 0$[/tex].

The nth derivative of (f(x)) can be expressed as [tex]$f^{(n)}(x) = n! \cdot a_n$.[/tex]Now, the (n+1)st derivative, [tex]$f^{(n+1)}(x)$[/tex], will be zero if and only if the leading coefficient [tex]$a_n = 0$[/tex]. However, this condition is not satisfied in general, as [tex]$a_n$[/tex] is assumed to be nonzero for a nontrivial polynomial. Therefore, the (n+1)st derivative is not guaranteed to be zero for all nth degree polynomials.

In mathematical terms,[tex]$F^{(n+1)}(x)$[/tex] equals zero if and only if the leading coefficient of (f(x)) is zero, but this is not a universal characteristic of nth degree polynomials. Consequently, the statement is false, and the (n+1)st derivative may not be zero for all x in the domain of the polynomial.

find the binomial coefficient: 2012/2011

Answers

²⁰¹²C₂₀₁₁ = (2012)! / [(2011)! (2012-2011)!]

²⁰¹²C₂₀₁₁ = (2012)! / [(2011)! (1)!]

Simplify 2012! / (2011)! = 2012

²⁰¹²C₂₀₁₁ = (2012)! / (1)! = 2012

Final answer:

The binomial coefficient '2012 choose 2011' is calculated using the formula C(n,k) = n! / [(n-k)! * k!]. After substituting the respective values into the formula, we find that the binomial coefficient of '2012 choose 2011' is 2012.

Explanation:

The binomial coefficient, often referred to in Mathematics, is generally expressed as 'n choose k' and calculated using the formula: C(n,k) = n! / [(n-k)! * k!]. In this formula, the '!' denotes factorial which means the product of an integer and all the integers below it.

However, the student's question seems to be asking for the binomial coefficient of '2012 choose 2011', which is misinterpreted as a fraction instead.

To calculate it correctly, we would apply the formula mentioned before: C(2012,2011) = 2012! / [(2012-2011)! * 2011!]. Because 2012-2011 equals 1, this simplifies our calculation. The factorial of 1 is 1 itself. Thus, C(2012,2011) = 2012!/ (1! * 2011!), which simplifies to be 2012. So the binomial coefficient of '2012 choose 2011' is 2012.

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A pile of dirt is cone shaped and it has a height of 10 feet and a diameter of 24 feet. Find the volume. * The answer is NOT 1570.96 or 1570.2*

Answers

You'll need to specify the accuracy to which you want your answer. For example: "Find the volume of this cone to 3 decimal place accuracy."

The formula for the volume of a cone is V = (1/3) (base) (height), where "base" represents the area of the base.

Here the diameter of the base is 24 feet, so the radius of the base is 12 feet.
Thus, the area of the base is (12 feet)^2 times pi: A = 144 pi ft^2.

Multiply this area by the height of the cone, which is 10 feet:

V = 1440 pi ft^3

By calculator, this volume is 4523.893421 cubic feet.

If you want this volume to the nearest cubic foot, it'd be 4524 cubic ft.
If you want this vol. to the nearest 100th cubic ft., it'd be 4523.89 cu. ft.

Let f(x) = -20x2 + 14x + 12 and g(x) =5x-6 Find f/g and state its domain a. 5x - 6; all real numbers except x =6/5 b. 5x - 6; all real numbers c. –4x – 2; all real numbers except x =6/5 d. –4x – 2; all real numbers

Answers

Final answer:

To find f/g, divide each term in f(x) by g(x). Resulting in f(x)/g(x) = -4x - 2 with the domain being all real numbers except x = 6/5. Hence, the correct answer is c. -4x - 2; all real numbers except x = 6/5.

Explanation:

To find the function f/g, we divide the function f(x) by g(x). Given f(x) = -20x2 + 14x + 12 and g(x) = 5x - 6, we divide these to get:

f(x)/g(x) = (-20x2 + 14x + 12) / (5x - 6)

Dividing each term in f(x) by g(x):

f(x)/g(x) = -4x - 2

The domain of this function would be all real numbers except where g(x) = 0, since we cannot divide by zero. g(x) = 0 when x = 6/5. Thus, the domain is all real numbers except x = 6/5.

The correct answer to the student's question is therefore c. -4x - 2; all real numbers except x = 6/5.

A chemist has three different acid solutions. The first acid solution contains
25%
acid, the second contains
40%
and the third contains
60%
. He wants to use all three solutions to obtain a mixture of
60
liters containing
45%
acid, using
3
times as much of the
60%
solution as the
40%
solution. How many liters of each solution should be used?

Answers

Let t, f, and s be the amount of 25%, 40%, and 60% solutions used.

We are told s=3f. The general equation is:

100(0.25t+0.4f+0.6s)/(t+s+f)=45, and using s=3f we have:

(0.25t+0.4f+1.8f)/(t+4f)=0.45

(0.25t+2.2f)/(t+4f)=0.45

0.25t+2.2f=0.45t+1.8f

0.25t+0.4f=0.45t

0.4f=0.2t

t=2f, remembering that s=3f, will allow us to solve for f

t+s+f=60, using t and s from above

2f+3f+f=60

6f=60

f=10 L then

s=3f=30 L and t=2f=20L

So 20L of 25%, 10L of 40%, and 30L of 60% acid solutions need to be mixed to make 60L of 45% acid solution.

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