15. The boss hands the letters to be printed to the secretary, one at a time, and places each on top of the pile. The secretary, whenever free, takes the top letter to type. One day, there are 9 letters to be typed, and the boss hands them over in the order of the first, second, $\cdots$, to the ninth. During lunch, the secretary tells a colleague that the eighth letter has been typed, but does not reveal any other details about the morning's work. The colleague is very curious to know which letters are left to be typed and in what order they will be printed.

Based on the above information, how many possible sequences are there for the letters typed in the morning? (Not typing any letter is also a possibility).
(Translated by Zhang Bingyi, proofread by Hou Zixin)
15. ( 704$)$

We consider two cases.
The first case: The ninth letter is finally typed by the secretary before noon.

This may be the order of the subset of the set $T=\{1,2$, ..., 6, 7, 9\} of possible subsets. In fact, the secretary can type a letter as soon as it arrives if its number is not in the subset, and not type other letters. $T$ has 8 elements, so there are $2^{8}=256$ subsets (including the empty set).

The second case: The ninth letter arrives at the secretary's office after noon.

Now the question is where this ninth letter will be "inserted" among the others? For each subset of $v=\{1,2, \cdots, 6,7\}$, each position is possible. For example, if the letters left at noon are 6, 3, 2, and the boss delivers the 9th letter as soon as the secretary finishes typing, then the typing sequence 6, 3, 9, 2 will occur.
For a permutation of $k$ letters, there are $k+1$ positions where the letter can be placed, before the secretary starts working, the boss delivers the 9th letter,
$$
=448 \text {. }
$$

Therefore, there are a total of $256+448=704$ possible typing sequences.