Let $$U = \{ 1,2,....,n\} $$, where n is a large positive integer greater than 1000. Let k be a positive integer less than n. Let A, B be subsets of U with $$|A| = |B| = k$$ and $$A \cap B = \phi $$. We say that a permutation of U separates A from B if one of the following is true.

- All members of A appear in the permutation before any of the members of B.

- All members of B appear in the permutation before any of the members of A.

How many permutations of U separate A from B?

The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is _______.

The coefficient of $${x^{12}}$$ in $${\left( {{x^3} + {x^4} + {x^5} + {x^6} + ...} \right)^3}\,\,\,\,\,\,$$ is _____________.

The number of distinct positive integral factors of 2014 is _______ .