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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 ∩ B = ∅. 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?

1. n!
2. (n choose 2k) (n - 2k)!
3. (n choose 2k) (n - 2k)! (k!)²
4. 2 (n choose 2k) (n - 2k)! (k!)²

Correct answer: 2 (n choose 2k) (n - 2k)! (k!)²

Solution

The correct option accounts for the selection of 2k elements from n, arranging the k elements of A and B in two distinct blocks (A before B and B before A), which introduces a factor of 2 for the two possible arrangements. The term (n choose 2k) represents the ways to choose the elements for A and B, while (n - 2k)! and (k!)² account for the arrangements of the remaining elements and the internal arrangements of A and B, respectively.

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