$I^{3} (m - n)^{3} + m^{3} (n - 1)^{3} + n^{3} (I - m)^{3}$ equals to ______.

(I I m n)^{3} (I - m) (m - n) (n - 1)

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(I m + m n + n l) (3 I^{2} M^{2} N^{2})

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3 I m n (m - n) (n - 1) (I - m)

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none of these

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Solution

The correct option is C none of these
$I^{3} (m - n)^{3} + m^{3} (n - 1)^{3} + n^{3} (I - m)^{3} = I^{3} [(m)^{3} - (n)^{3} - I (m n) (m - n)] + m^{3} (n)^{3} - (1)^{3} - 3 (x) (x - 1) + n^{3} (I)^{3} - (m)^{3} (m I) (I - m) = I^{3} [m^{3} - n^{3} - 3 m n (m - n)] + m^{3} [n^{3} - 1 - 3 n (n - 1)] + n^{3} [I^{3} - m^{3} - 3 m I (I - m)] = m^{3} I^{3} - n^{3} I^{3} - 3 m^{2} n I^{3} + 3 m n^{2} I^{2} + m^{3} n^{3} - m^{3} - 3 n^{2} m^{3} + 3 n m^{3} + n^{3} I^{3} - n m^{3} - 3 m I^{2} n^{3} + 3 m^{2} n^{3}$

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