Two numbers ‘a’ & ‘b’ are chosen from the set of {1,2,3……3n}. In how many ways can these integers be selected such that $a^{2} - b^{2}$ is divisible by 3

$\frac{3}{2} n (n + 1) + n^{2}$

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$\frac{3}{2} n (n - 1) + n^{2}$

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$\frac{1}{2} n (n + 1) - n^{2}$

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$\frac{1}{2} n (n - 1) + n^{2}$

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Solution

The correct option is B
$\frac{3}{2} n (n - 1) + n^{2}$

$G_{1} : 3, 6, 9......3 n G_{2} : 1, 4, 7..... (3 n - 2) G_{3} : 2, 5, 8.... (3 n - 1) a^{2} - b^{2} = (a - b) (a + b)$
Either a-b is divisible by 3 (or) a + b is divisible by 3 (or) both
$n c_{2} + n c_{2} + n c_{2} + n c_{1} . n c_{1} 3 \frac{n (n - 1)}{2} + n^{2}$

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Two numbers ‘a’ & ‘b’ are chosen from the set of {1,2,3……3n}. In how many ways can these integers be selected such that a2−b2 is divisible by 3

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Two numbers ‘a’ & ‘b’ are chosen from the set of {1,2,3……3n}. In how many ways can these integers be selected such that $a^{2} - b^{2}$ is divisible by 3