1. Given the problem: If $\beta(m,n) = 6$ and $\Gamma(m,n) = 120$, find $\Gamma_m \Gamma_n$. 2. We need to understand the relationship between $\beta(m,n)$, $\Gamma(m,n)$, and $\Gamma_m \Gamma_n$. Typically, $\beta(m,n)$ is the Beta function and $\Gamma(m,n)$ is the Gamma function. 3. The Beta function is defined as: $$\beta(m,n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}$$ 4. Given $\beta(m,n) = 6$ and $\Gamma(m,n) = 120$, we interpret $\Gamma(m,n)$ as $\Gamma(m+n) = 120$. 5. Substitute the known values into the Beta function formula: $$6 = \frac{\Gamma(m) \Gamma(n)}{120}$$ 6. Multiply both sides by 120: $$\Gamma(m) \Gamma(n) = 6 \times 120 = 720$$ 7. Therefore, the value of $\Gamma_m \Gamma_n$ is $720$. Final answer: 720