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Proabability

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Two integers are selected from the first 1024 positive integral perfect squares. What is the probability that both of these numbers are fifth powers of integers?

Sep 27, 2022

#1
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Firstly, it would be polite to respond after each time someone answers a question. Especially when it is a 'real' answer.

Sep 27, 2022
#2
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You did it again Melody. Called someone out for something not true, and provided a non-workable solution. All solutions on google require for you to pay.

Sep 28, 2022
#3
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This is a post I was referring to.  It is rude to not respond and then just repost.

https://web2.0calc.com/questions/probability_37639#r1

Yes, the second part of my comment may not be correct. I appologise for that.

Melody  Sep 28, 2022
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1024^2 = 1048576

1048576^(1/5) = 16

4/1024  * 3/1023 =  12/1047552 = 3/261888

EDIT:  It is only my last disision that is incorrect    12 /1047552 =  1/87296  NOW our answers agree

Thanks Builderboi.    Oh, and your answer is a lot more elegent than mine is

Sep 28, 2022
edited by Melody  Sep 29, 2022
edited by Melody  Sep 29, 2022
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I got a different answer, Melody.

If a number is both a square and a 5th power, it must also be a perfect 10th power.

Note that $$1024^2 = 4^{10}$$, so there are 4 numbers that work: $$1^{10}$$$$2^{10}$$$$3^{10}$$, and $$4^{10}$$

There are $${4 \choose 2} = 6$$ successes and $${1024 \choose 2} = 523776$$ total outcomes.

So, the probability is $${6 \over 523776} = \color{brown}\boxed{1 \over 87296}$$

Sep 29, 2022