Limiting factorials

Calculus Level 4

\[ \large \lim_{ n \to\infty }{ \left( \frac{ 5n \choose 3n }{ 3n \choose 2n } \right) }^{1/n} =\frac{ { a }^{ a } }{ { b }^{ 2b } } \]

The equation above holds true for positive integers \(a\) and \(b\). Find the value of \( a+b \).

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Bonus: Do it without Stirling's approximation.

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