Consider the following problem:
Solve the inequality $\left( \frac { 1 } { 4 } \right) ^ { n } < \left( \frac { 1 } { 32 } \right) ^ { 10 }$, where $n$ is a positive integer.
A student produces the following argument:
$$\begin{array} { r l r } \left( \frac { 1 } { 4 } \right) ^ { n } & < \left( \frac { 1 } { 32 } \right) ^ { 10 } & \\ \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) ^ { n } & < \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) ^ { 10 } & ( \mathrm { I } ) \\ n \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) & < 10 \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) & \downarrow ( \mathrm { II } ) \\ n < \frac { 10 \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 32 } \right) } { \log _ { \frac { 1 } { 2 } } \left( \frac { 1 } { 4 } \right) } & \downarrow ( \mathrm { III } ) \\ n < \frac { 10 \times 5 } { 2 } = 25 & \downarrow ( \mathrm { IV } ) \\ 1 \leqslant n \leqslant 24 & \downarrow ( \mathrm {~V} ) \end{array}$$
Which step (if any) in the argument is invalid?
A There are no invalid steps; the argument is correct
B Only step (I) is invalid; the rest are correct
C Only step (II) is invalid; the rest are correct
D Only step (III) is invalid; the rest are correct
E Only step (IV) is invalid; the rest are correct
F Only step (V) is invalid; the rest are correct