Catalan Numbers

Source: http://milan.milanovic.org/math/english/fibo/fibo4.html

It starts by asking a strange-sounding question:

In how many ways can a collection of parentheses appear in a sentence?

http://en.wikipedia.org/wiki/Catalan_number

http://mathworld.wolfram.com/CatalanNumber.html

http://geometer.org/mathcircles/catalan.pdf

http://mathforum.org/advanced/robertd/catalan.html

(1)

In how many ways can you cut up a regular hexagon (6-sided polygon) into triangles using 3 nonintersecting diagonals?
For example, there are 5 ways for a pentagon

http://www.math.utah.edu/mathcircle/notes/mladen2.pdf

(2)

Let's say 20 people show up to go see the $5 matinee at the local theater.
Suppose that 10 of these people have exact change, but the other 10 have only a $10 bill,and will require change. Unfortunately, on this particular day the cashier has forgotten to stop at the bank and so he has no change to start with. What are the odds that he will be able to sell all 20 people tickets without running out of change?

http://www.math.harvard.edu/~mantovan/ADMIN/catalan2.pdf