Combination

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?

There are 6 jobs with distinct difficulty levels, and 3 computers with distinct processing speeds. Each job is assigned to a computer such that:

The fastest computer gets the toughest job and the slowest computer gets the easiest job.
Every computer gets at least one job.

The number of ways in which this can be done is ___________.

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 number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ______________.

The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3} is____________.