Given that $2^{2004}$ is a $604$-[digit](https://artofproblemsolving.com/wiki/index.php/Digit) number whose first digit is $1$, how many [elements](https://artofproblemsolving.com/wiki/index.php/Element) of the [set](https://artofproblemsolving.com/wiki/index.php/Set) $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ have a first digit of $4$? 
$\mathrm{(A)}\ 194 \qquad \mathrm{(B)}\ 195 \qquad \mathrm{(C)}\ 196 \qquad \mathrm{(D)}\ 197 \qquad \mathrm{(E)}\ 198$
Given $n$ digits, there must be exactly one power of $2$ with $n$ digits such that the first digit is $1$. Thus $S$ contains $603$ elements with a first digit of $1$. For each number in the form of $2^k$ such that its first digit is $1$, then $2^{k+1}$ must either have a first digit of $2$ or $3$, and $2^{k+2}$ must have a first digit of $4,5,6,7$. Thus there are also $603$ numbers with first digit $\{2,3\}$ and $603$ numbers with first digit $\{4,5,6,7\}$. By using [complementary counting](https://artofproblemsolving.com/wiki/index.php/Complementary_counting), there are $2004 - 3 \times 603 = 195$ elements of $S$ with a first digit of $\{8,9\}$. Now, $2^k$ has a first digit of $\{8,9\}$ [if and only if](https://artofproblemsolving.com/wiki/index.php/Iff) the first digit of $2^{k-1}$ is $4$, so there are $\boxed{195} \Rightarrow \mathrm{(B)}$ elements of $S$ with a first digit of $4$.