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:
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
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.