Binomial + Trigo + Complex!

Algebra Level 5

\[^n C_0 - \frac {^n C_2}{ (2 + \sqrt 3)^2} + \frac {^n C_4}{ (2 + \sqrt 3)^4} - \frac {^n C_6}{ (2 + \sqrt 3)^6} + \ldots + \frac{^n C_n}{(2+\sqrt 3)^n}\]

Let \(^a C_b \) denote the binomial coefficient \( {a \choose b} \), and for any positive integer \(n=12m \), the expression above equals to \(\displaystyle (-1)^m \left ( \frac{ \sqrt P}{ 1 + \sqrt Q} \right )^n \), where \(P\) and \(Q\) are positive integers.

Find the value of \(P-Q\).

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