JEE Main 2025 — 3 April, Morning Shift — Question 20

$\mid 2$ adj $(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A})) \mid$

$2^{3} .|3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A})|^{2}$

$2^{3} \cdot\left(3^{3}\right)^{2} \cdot|\mathrm{~A}|^{2} \cdot|\operatorname{adj}(2 \mathrm{~A})|^{2}$

$2^{3} \cdot 3^{6} \cdot|\mathrm{~A}|^{2} \cdot\left(|2 \mathrm{~A}|^{2}\right)^{2}$

$2^{3} \cdot 3^{6} \cdot|\mathrm{~A}|^{2}\left[\left(2^{3}\right)^{2} \cdot|\mathrm{~A}|^{2}\right]^{2}$

$2^{3} \cdot 3^{6} \cdot|\mathrm{~A}|^{2} \cdot 2^{12} \cdot|\mathrm{~A}|^{4}$

$2^{15} .3^{6} .|\mathrm{A}|^{6}$

$2^{15} \cdot 3^{6} \cdot 5^{6}=2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}$

$\alpha=15, \quad \beta=6, \quad \gamma=6$

$\alpha+\beta+\gamma=27$