Mathematics 501
Homework (due Sep 20)
A. Hulpke
16) a) Show that the number of different mountain ranges you can draw with n upstrokes and n downstrokes is given by the Catalan number Cn+1 :
1
1
2
5
b) A clown stands at the edge of a swimming pool with a bowl with n red and n blue balls. He randomly
draws balls from it. If he draws a red ball he takes a step back. If he draws a blue ball he takes a step
forward. Assuming that the steps are always the same length, what is the probability that the clown stays
dry?
17) Show that there are Cn+1 rooted trees (fixing a “root” vertex) with n edges when distinguishing left
and right branches:
1
1
2
5
18) Show that (ignoring equivalences by symmetry) there are Cn different ways to divide a polygon with
n + 1 sides into triangular regions by inserting diagonals that do not intersect in the interior.
19) Determine thee number of ways to color the squares of a 1 × n chessboard using red, yellow and green,
where the number of red squares is even and there is at least one green square.
20∗ ) Consider the number of different ways to fold a strip of n stamps, disregarding the distinction of
front and back:
How many different possibilities are there to fold a strip of five stamps?
Problems marked with ∗ are bonus problems for extra credit.