1.58 The manager hands the letters to be printed to the secretary one by one, and always places them on top of the pile. The secretary, whenever free, takes the topmost letter to print. One day, there are 9 letters to be printed, and the manager hands them over in the order of the 1st, 2nd, ..., 9th letter. At lunchtime, the secretary tells a colleague that the 8th letter has been printed, but does not reveal any other details about the morning's work. The colleague is curious to know in what order the letters were printed, and how many possible orders there are for printing the letters in the afternoon (including the possibility of no letters left to print).
[Solution] We count the possibilities in two separate cases.
(1) The 9th letter is delivered to the secretary in the morning. Let
$$
T=\{1,2,3,4,5,6,7,9\},
$$

then each possible set of letters printed in the afternoon is a subset of $T$, because the secretary could have finished printing the letters not in the subset as soon as they were delivered in the morning, without printing any others. Set $T$ has 8 elements, so it has $2^{8}=256$ different subsets (including the empty set).
(2) The 9th letter is delivered to the secretary in the afternoon. Let
$$
S=\{1,2,3,4,5,6,7\},
$$

then the numbers of letters not printed in the morning are a subset of $S$. If we place 9 after the subset, the situation is the same as in (1), so the only new possible sequences are those in which at least one letter has been printed in the afternoon before the 9th letter is delivered. This is equivalent to placing the number 9 in a position other than the last in the subset. For a subset with $k$ elements, the number 9 has $k$ positions to be placed, i.e., between the $(i-1)$-th and the $i$-th element, $i=1,2, \cdots, k$. Thus, the total number of different sequences is
$$
\sum_{k=0}^{7} k C_{7}^{k}=\sum_{k=1}^{7} 7 C_{6}^{k-1}=7 \times 2^{6}=448,
$$

i.e., there are 448 possible printing sequences in the afternoon.