Get your brackets straight
In an attempt to fix mis-matched parenthesis you attempt all the possible configuration, but you quickly realize how messy it is.
How many distinct valid parentheses permutations are possible for 50 expressions?
Details and Assumptions
As an explicit example the number of valid permutations of 4 parentheses is 2:
(( ))
and
()()
.
All the opening parentheses have a matching closing parentheses.
The answer is 4861946401452.
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1 solution
View a list of parentheses as a walk on a square 25x25 grid, where left parentheses correspond to rightward steps and right parentheses correspond to upward steps. Balanced lists of parentheses are in exact correspondence with walks from the lower left to upper right corners that stay below the diagonal. The number of such walks is the 25th Catalan number, 4 8 6 1 9 4 6 4 0 1 4 5 2 .