Greatest term!

Algebra Level 3

Consider the binomial expansion \( (2+3x)^9 \). Let \( \mathbb{P} \) denote the largest coefficient of all terms when it's expanded.

If \( \left ( \dfrac {3}{2} \right )^6 \times \mathbb{P} = \dfrac {3^a \times b}{c} \)

for natural numbers \(a,b,c\) with \(\text{gcd}(c,3)=\text{gcd}(b,c) = 1 \).

What is the value of \(a+b+c\)?

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