11. Let n be an integer, . Let denote the number of subsets of whose order is congruent to for . Determine in terms of n.

12. Prove by mathematical induction that for

13. Show that the number of distinct ways of triangulating a convex n-gon by nonintersecting diagonals equals .

14. Show that the number of solutions of the equation in positive integers is .

15. Show that for each n and k, the following is true:

16. Calculate

17. Calculate when t is a real number.

18. Let k be a non-negative integer number. Show that any non-negative integer number n can be written uniquely as

where .

19. Let denote the nth Bell number. Show that for each .

20. Determine the number of ways of writing a positive integer n as a sum of ones and twos.

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Nalin Pithwa