How many sequences in a prefix code can be compressed by m bits?
There is no problem with the statement. An encoding $f$ of the set of strings of length $n$ is a map from $\{0,1\}^n$ to $\{0,1\}^*$ which is injective (one-to-one). Such a mapping compresses a string $x$ by more than $m$ if $|f(x)| &lt; |x| - m$. In our case, all the strings have length $n$, so a string $x \in \{0,1\}^n$ is compressed by more than $m$ if $|f(x)| &lt; n-m$. There are $\sum_{i=0}^{n-m-1} 2^i = 2^{n-m}-1 &lt; 2^{n-m}$ such strings, and since $f$ is injective, the fraction of strings compressed by more than $m$ is less than $2^{n-m}/2^n = 2^{-m}$.