Catalan numbers can be defined in lots of ways. One of those ways is: (nth Catalan number) Cn counts the number of expressions containing n pairs of parentheses which are correctly matched: for example for n = 3, C_n = 5 ((())) ()(()) ()()() (())() (()())
This wiki article gives a nice formula for $C_n$:
C_n = ((2n)!/(n!n!)) / (n+1) where n = number of pairs of parentheses. I wanted to figure out the value of C_n by finding all balanced expressions by n pairs of parentheses. So, we have 2n parentheses which creates (2n)! arrangements out of which we have n identical open parentheses and n closed parentheses so we divide by n!*n!. Total unique arrangements would be (2n)!/(n!n!) but not all of them are balanced. So how does dividing by n+1 solves the problem?