2004 AMC 12B Problems/Problem 25

Problem

Given that $2^{2004}$ is a $604$ -digit number whose first digit is $1$ , how many elements of the set $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ have a first digit of $4$ ?

Given $n$ digits, there must be a 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 either $\{2,3\}$ or $\{4,5,6,7\}$ . By the complement principle, there are $2004 - 3 \times 603 = 195$ elements of $S$ with a first digit of $\{8,9\}$ . Now, $2^k$ has a first of $\{8,9\}$ if and only if 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 $4$ .