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The correct answer is $20$. It’s given that $f\left(x\right)=-a^x+b$. Substituting $-a^x+b$ for $f\left(x\right)$ in the equation $y=f\left(x\right)-12$ yields $y=-a^x+b-12$. It’s given that the y-intercept of the graph of $y=f\left(x\right)-12$ is $\left(0,-\frac{75}{7}\right)$. Substituting $0$ for $x$ and $-\frac{75}{7}$ for $y$ in the equation $y=-a^x+b-12$ yields $-\frac{75}{7}=-a^0+b-12$, which is equivalent to $-\frac{75}{7}=-1+b-12$, or $-\frac{75}{7}=b-13$. Adding $13$ to both sides of this equation yields $\frac{16}{7}=b$. It’s given that the product of $a$ and $b$ is $\frac{320}{7}$, or $ab=\frac{320}{7}$. Substituting $\frac{16}{7}$ for $b$ in this equation yields $\left(a\right)\left(\frac{16}{7}\right)=\frac{320}{7}$. Dividing both sides of this equation by $\frac{16}{7}$ yields $a=20$.