If $a = 2012, b = 2011, c = 2010$ then the values $a^{2} + b^{2} + c^{2} - a b - b c - c a$ is:

0

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2012

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3

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4024

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Solution

The correct option is C $3$
Given,
$a = 2012, b = 2011, c = 2010$
Let's say
$X = a^{2} + b^{2} + c^{2} - a b - b c - c a$

So, $2 X = 2 (a^{2} + b^{2} + c^{2} - a b - b c - c a)$

$⟹ 2 X = 2 a^{2} + 2 b^{2} + 2 c^{2} - 2 a b - 2 b c - 2 c a$

$⟹ 2 X = a^{2} - 2 a b + b^{2} + b^{2} - 2 b c + c^{2} + c^{2} - 2 c a + a^{2}$

$⟹ 2 X = (a - b)^{2} + (b - c)^{2} + (c - a)^{2}$

$⟹ 2 X = 1 + 1 + 4$

$⟹ 2 X = 6 ⟹ x =$ 3

Hence the value of $a^{2} + b^{2} + c^{2} - a b - a c - b c$ is $3$ .

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a^{2} + b^{2} + c^{2} - a b - b c - c a

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