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.
More later,
Nalin Pithwa