1. Find the number of integers between 1 and 10, 000 inclusive that are not divisible by 4, 6, 7, or 10

2. etermine the number of 10-combinations of the multiset S = { a, 4 b, 5 c, 7 d} (this means the multiset S has the element a repeated infinitely many times, the element b with repetition number 4, the element c with repetition number 5, and the element d with repetition number 7).

3. bakery sells chocolate, cinnamon, and plain doughnuts and at a particular time has 6 chocolate, 6 cinnamon, and 3 plain. If a box contains 12 doughnuts, how many different options are there for a box of doughnuts?

4. Determine the number of solutions of the equation x1+x2+x3+x4+x5=14

in positive integers x1, x2, x3, x4, and x5, with every variable xj not exceeding 5.

5. Determine the number of permutations of {1, 2, . . . , 9} in which at least one odd integer is in its natural position (the "natural position" of an integer j is the jth position from the left)

6. Determine a general formula for the number of permutations of the set {1, 2, . . . , n} in which exactly k integers are in their natural positions. (You may express your answer in terms of the number of derangements Dm for a certain integer m.)