# 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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