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dlinn [17]
2 years ago
8

) 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
Mathematics
1 answer:
Brilliant_brown [7]2 years ago
8 0
<span>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!. 


</span>
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What’s the answer? Because am having a hard time with this math problem.
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9514 1404 393

Answer:

  34.5 square meters

Step-by-step explanation:

We assume you want to find the area of the shaded region. (The actual question is not visible here.)

The area of the triangle (including the rectangle) is given by the formula ...

  A = 1/2bh

The figure shows the base of the triangle is 11 m, and the height is 1+5+3 = 9 m. So, the triangle area is ...

  A = (1/2)(11 m)(9 m) = 49.5 m^2

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The figure shows the rectangle is 5 m high and 3 m wide, so its area is ...

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The percentage of body fat of a random sample of 36 men aged 20 to 29 found a sample mean of 14.42. Find a 95% confidence interv
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Answer:

14.42-1.96\frac{6.95}{\sqrt{36}}=12.150    

14.42+ 1.96\frac{6.95}{\sqrt{36}}=16.690    

So on this case the 95% confidence interval would be given by (12.150;16.690)    

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

\bar X=14.42 represent the sample mean

\mu population mean (variable of interest)

\sigma=6.95 represent the population standard deviation

n=36 represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}   (1)

Since the Confidence is 0.95 or 95%, the value of \alpha=0.05 and \alpha/2 =0.025, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that z_{\alpha/2}=1.96

Now we have everything in order to replace into formula (1):

14.42-1.96\frac{6.95}{\sqrt{36}}=12.150    

14.42+ 1.96\frac{6.95}{\sqrt{36}}=16.690    

So on this case the 95% confidence interval would be given by (12.150;16.690)    

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