The problem of finding the number of different ways in which a product of 
different ordered factors can be calculated by pairs (i.e.,
the number of binary bracketings of 
letters). For example, for the four factors 
, 
, 
, and 
, there are five possibilities: 
, 
, 
, 
, and 
.
The solution was given by Catalan in 1838 as
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is a multifactorial, 
is the usual factorial, and
is a so-called Catalan
number.
