combinations

[donys]

Hi guys.

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Can you help me with these problems in combinations.I dont know if my solutions are correct.

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Problem 1

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There are 3 classes,each one with n students.We must select 3 students from the total of 3n students.

i)How many are the possible choices?

C=combinations

The possible choices are:

C(n,3)*C(n,0)*C(n,0)*3 + C(n,2)*C(n,1)*C(n,0)*3! + C(n,1)*C(n,1)*C(n,1)

 

ii)How many are the possible choices,if the 3 students belong to the same class?

C(n,3)*C(n,0)*C(n,0)*3

 

iii)How many are the possible choices,if 2 of the 3 students belong to the same class and the third to a different class?

C(n,2)*C(n,1)*C(n,0)*3!

 

iv)How many are the possible choices,if the 3 students belong to a different class?

C(n,1)*C(n,1)*C(n,1)

 

v)Use your answers i)-iv) to expand C(3n,n)

I dont know this one

 

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Problem 2

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A number of n students attend the lecture of probabilities.The array with the results of the final exam includes only the names of the students that passed the lesson in descending order.We assume that there arent any students with the same grade.How many are the possible arrays?

C(n,1)+C(n,2)*2!+C(n,3)*3!+...+C(n,n)n!

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Problem 3

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We put the hats of n persons in a box.Then each person randomly chooses a hat from the box.Find the probability that n-2 persons

will accurately choose their own hat.

 

I guessed that n-2 persons choose acurately their own hat is the same

with: 1 - 2 persons dont take their own hat

=[1 - C(n,1)*(n-1)!+C(n-1,1)*(n-2)!] / n!

=[1-n!+(n-1)!]n!?????i'm not sure??????

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