For $f_n(x)=x^n$ and $a\neq 1$ consider 

I. $(f_{11}(a)f_{13}(a))^{14}$

II. $f_{11}(a)f_{13}(a)f_{14}(a)$

III. $(f_{11}(f_{13}(a)))^{14}$

IV. $f_{11}(f_{13}(f_{14}(a)))$

Which of these equal $f_{2002}(a)$?

$\textbf{(A) }\text{I and II only}\qquad\textbf{(B) }\text{II and III only}\qquad\textbf{(C) }\text{III and IV only}\qquad\textbf{(D) }\text{II, III, and IV only}\qquad\textbf{(E) }\text{all of them}$
To determine which of the given expressions equal \( f_{2002}(a) \), we need to simplify each expression using the definition \( f_n(x) = x^n \).

1. **Expression I: \( (f_{11}(a)f_{13}(a))^{14} \)**
   \[
   f_{11}(a) = a^{11} \quad \text{and} \quad f_{13}(a) = a^{13}
   \]
   \[
   f_{11}(a)f_{13}(a) = a^{11} \cdot a^{13} = a^{11+13} = a^{24}
   \]
   \[
   (f_{11}(a)f_{13}(a))^{14} = (a^{24})^{14} = a^{24 \cdot 14} = a^{336}
   \]
   Therefore, Expression I simplifies to \( f_{336}(a) \).

2. **Expression II: \( f_{11}(a)f_{13}(a)f_{14}(a) \)**
   \[
   f_{11}(a) = a^{11}, \quad f_{13}(a) = a^{13}, \quad \text{and} \quad f_{14}(a) = a^{14}
   \]
   \[
   f_{11}(a)f_{13}(a)f_{14}(a) = a^{11} \cdot a^{13} \cdot a^{14} = a^{11+13+14} = a^{38}
   \]
   Therefore, Expression II simplifies to \( f_{38}(a) \).

3. **Expression III: \( (f_{11}(f_{13}(a)))^{14} \)**
   \[
   f_{13}(a) = a^{13}
   \]
   \[
   f_{11}(f_{13}(a)) = f_{11}(a^{13}) = (a^{13})^{11} = a^{13 \cdot 11} = a^{143}
   \]
   \[
   (f_{11}(f_{13}(a)))^{14} = (a^{143})^{14} = a^{143 \cdot 14} = a^{2002}
   \]
   Therefore, Expression III simplifies to \( f_{2002}(a) \).

4. **Expression IV: \( f_{11}(f_{13}(f_{14}(a))) \)**
   \[
   f_{14}(a) = a^{14}
   \]
   \[
   f_{13}(f_{14}(a)) = f_{13}(a^{14}) = (a^{14})^{13} = a^{14 \cdot 13} = a^{182}
   \]
   \[
   f_{11}(f_{13}(f_{14}(a))) = f_{11}(a^{182}) = (a^{182})^{11} = a^{182 \cdot 11} = a^{2002}
   \]
   Therefore, Expression IV simplifies to \( f_{2002}(a) \).

From the above simplifications, we see that:
- Expression I simplifies to \( f_{336}(a) \)
- Expression II simplifies to \( f_{38}(a) \)
- Expression III simplifies to \( f_{2002}(a) \)
- Expression IV simplifies to \( f_{2002}(a) \)

Thus, the correct answer is that Expressions III and IV equal \( f_{2002}(a) \).

The final answer is \( \boxed{ \textbf{(C) }\text{III and IV only} } \)