Catalan's Problem

The problem of finding the number of different ways in which a Product of $n$ different ordered Factors can be calculated by pairs (i.e., the number of Binary Bracketings of $n$ letters). For example, for the four Factors $a$, $b$, $c$, and $d$, there are five possibilities: $((ab)c)d$, $(a(bc))d$, $(ab)(cd)$, $a((bc)d)$, and $a(b(cd))$. The solution was given by Catalan in 1838 as

$$C_n' = {2\cdot 6\cdot 10\cdot (4n-6)\over n!}$$

and is equal to the Catalan Number $C_{n-1}=C_n'$.

See also Binary Bracketing, Catalan's Diophantine Problem, Euler's Polygon Division Problem

References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 23, 1965.