# Mathematical Expressions

Probability Level 5

When writing a math expression, any time there is an open bracket "$($", it is eventually followed by a closed bracket "$)$". When we have a complicated expression, there may be several brackets nested amongst each other, such as in the expression $(x+1)*((x-2) + 3(x-4)\times(x^2 + 7\times(3x + 4)))$. If we removed all the symbols other than the brackers from the expression, we would be left with the arrangement $()(()()(())).$ For any arrangement of brackets, it could have come from a valid mathematical expression if and only if for every place in the sequence, the number of open brackets before that place is at least as large as the number of closed brackets. If $34$ open brackets and $34$ closed brackets are randomly arranged, the probability that the resulting arrangement could have come from a valid mathematical expression can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $a + b$?

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