Problem (combinatorics)

Provide a solution with justification for each of the following:

1.If repetitions are not allowed, how many 7-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7?

2.Find the number of ways in which p like jobs may be assigned to (q+1) persons, if no person is left unemployed and we have p>q. Hint: this problem is equivalent to that of counting the number bp,q of binary strings with no consecutive 1's, with a 0 at each end. with q 1's and p 0's. You should consider establishing a recursive formula for the numbers bp,q similar to that of the Pascal triangle.

3.There are 3 pigeon holes marked A, B, C. In how many ways can I arrange 10 different postcards so that 5 of them are in A, 3 in B and 2 in C?

4.Consider all bit strings of length 14. How many have exactly eleven 1's such that none of these 1's are adjacent to each other?