IIT JEE , AIEEE Forum - Mathematics , Algebra - probablity of the reverse psychology theory

Mr.IITIAN007

Two numbers X and Y are chosen at random( without replacement ) from the set
{1,2,3,4,..........................,5N).What is the probability that
(i) X³+Y³is divisible by 5
(ii) X²+Y² is divisible by 5

iitkgp_bipin

I II III IV V

1 2 3 4 5

6 7 8 9 10

........................................

5n-4 5n-3 5n-2 5n-1 5n

x³ + y³ is divisible by 5 if x + y is divisible by 5.

Hence x and y can be choosen from I and IV , II and III or both from V.

No. of ways of choosing x and y from I and IV = (n)(n) = n²

No. of ways of choosing x and y from II and III = (n)(n) = n²

No. of ways of choosing x and y from V = ⁿC₂ = n(n-1)/2

Total no. of required ways =n² + n² + n(n-1)/2 = n(5n-1)/2

Total no. of ways of choosing x and y from 5n numbers = ⁵ⁿC₂ = 5n(5n-1)/2

Required probability = {n(5n-1)/2}/{5n(5n-1)/2} = 1/5

Mr.IITIAN007

I agree with you sir.Thanks,but what about the second part of my query.

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