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Two numbers a and b are chosen at random from the set {1,2,3,..,5n}. The probability that `a^(4)-b^(4)` is divisible by 5, is

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Two numbers a and b are chosen at random from the set {1,2,3,..,5n}. The probability that a4-b4 is divisible by 5, is A. 17n-5 25n-1 B. 17n+5 5(5n-1) C. 17n-5 5(5n-1) D. none of these Select the correct answer from above options

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Correct Answer - C The number of ways of choosing a and b from the given set of 5n integers is .5nC2. Let us divide the given set of 5n integers in 5 groups as follows : G1:1,6,11,..,5n-4 G2:2,7,12,..,5n-3 G3:3,8,13,..,5n-2 G4:4,9,14,..,5n-1 G5:5,10,15,..,5n We have, a4-b4=(a-b)(a2+b2) So, we observe that a4-b4 will be divisible by 5, if both a and b belong to the last group or if they belong to any of the remaining four groups. Thus, the number of favourable elementary events is .nC2+.4nC2. Hence, required probability = .nC2+.4nC2 .5nC2= 17n-5 5(5n-1)

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