probabilty

two numbers 'a' and 'b' are selected at random from the set {1,2,3,4,5............3n}. find the probabilty that a²-b² is divisible by 3

1 Answers

arrange nos. like this........
1 4 7........3n-2
2 5 8............3n-1
3 6.........3n
now a²-b² can be written as (a+b)(a-b).

a-b is divisible by 3 if we select two nos. from a group.so no. of ways to select nos. are=3*ⁿC₂.

a+b is divisible by 3 only if nos. are selected from third group.so no. of ways are ⁿC₂.

so, probability=3*ⁿC₂.+ ⁿC₂ /³ⁿC₂

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