Problem: In how many ways can brackets be placed around a sequence of n + 1 letters so that there are two letters inside each pair of brackets?

Example: ab = (ab) = 1 way
Example: abc = (ab)c, a(bc) = 2 ways
Example: abcd = (ab)(cd), a((bc)d), ((ab)c)d, a(b(cd)), (a(bc))d = 5 ways

Extensions: Find a formula for the nth term.

Note: You will discover what is known as Catalan Numbers. This also can find how many ways a regular n-gon can be divided into n - 2 triangles if different orientations are counted separately. This is also the way for determining how many ways can n votes be cast between 2 candidates so that the one chosen candidate is never behind in the counting