1. **State the problem:** We need to compare $A = n$ and $B = 12$ given the equation $n^{11} = 11^n$. 2. **Analyze the equation:** The equation $n^{11} = 11^n$ means the number $n$ raised to the 11th power equals 11 raised to the $n$th power. 3. **Check if $n=11$ is a solution:** Substitute $n=11$: $$11^{11} = 11^{11}$$ This is true, so $n=11$ satisfies the equation. 4. **Compare $A$ and $B$ for $n=11$:** $A = 11$, $B = 12$, so $A < B$. 5. **Check if other $n$ values satisfy the equation:** - For $n=12$, $12^{11}$ vs $11^{12}$: Calculate approximate values: $12^{11} = 743008370688$ (approx) $11^{12} = 3138428376721$ (approx) Since $12^{11} < 11^{12}$, $n=12$ is not a solution. 6. **Behavior of the function:** Define $f(n) = n^{11} - 11^n$. - At $n=11$, $f(11)=0$. - At $n=12$, $f(12) < 0$. - At $n=10$, $10^{11} = 100000000000$, $11^{10} = 25937424601$, so $f(10) > 0$. This suggests $n=11$ is the only integer solution. 7. **Conclusion:** Since $n=11$ satisfies the equation and $A = n = 11 < 12 = B$, the answer is option (a) $A$. **Final answer:** a) A