In an office, at various times during the day the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. If there are five letters in all, and the boss delivers them in the order $1\  2\  3\  4\  5$, which of the following could [b]not[/b] be the order in which the secretary types them?

$ \textbf{(A)}\ 1\ 2\ 3\ 4\ 5  \qquad\textbf{(B)}\ 2\ 4\ 3\ 5\ 1 \qquad\textbf{(C)}\ 3\ 2\ 4\ 1\ 5 \qquad\textbf{(D)}\ 4\ 5\ 2\ 3\ 1 \\ \qquad\textbf{(E)}\ 5\ 4\ 3\ 2\ 1 $
To determine which order the secretary could **not** type the letters, we need to understand the behavior of a stack (LIFO - Last In, First Out). The secretary always types the letter that is on top of the pile. Let's analyze each option to see if it is possible under these conditions.

1. **Option (A): 1, 2, 3, 4, 5**
   - The boss delivers the letters in the order 1, 2, 3, 4, 5.
   - The secretary types them in the same order they were delivered.
   - This is possible because the secretary types each letter immediately after it is delivered.
   - Therefore, option (A) is possible.

2. **Option (B): 2, 4, 3, 5, 1**
   - The boss delivers the letters in the order 1, 2, 3, 4, 5.
   - The secretary types letter 2 first, which means letter 1 must still be in the pile.
   - Then, the secretary types letter 4, which means letters 3 and 1 must still be in the pile.
   - Next, the secretary types letter 3, which means letter 1 must still be in the pile.
   - Then, the secretary types letter 5, which means letter 1 must still be in the pile.
   - Finally, the secretary types letter 1.
   - This sequence is possible because the secretary can always type the top letter.
   - Therefore, option (B) is possible.

3. **Option (C): 3, 2, 4, 1, 5**
   - The boss delivers the letters in the order 1, 2, 3, 4, 5.
   - The secretary types letter 3 first, which means letters 1 and 2 must still be in the pile.
   - Then, the secretary types letter 2, which means letter 1 must still be in the pile.
   - Next, the secretary types letter 4, which means letter 1 must still be in the pile.
   - Then, the secretary types letter 1.
   - Finally, the secretary types letter 5.
   - This sequence is possible because the secretary can always type the top letter.
   - Therefore, option (C) is possible.

4. **Option (D): 4, 5, 2, 3, 1**
   - The boss delivers the letters in the order 1, 2, 3, 4, 5.
   - The secretary types letter 4 first, which means letters 1, 2, and 3 must still be in the pile.
   - Then, the secretary types letter 5, which means letters 1, 2, and 3 must still be in the pile.
   - At this point, the secretary cannot type letter 2 because letter 3 is on top of it.
   - Therefore, option (D) is not possible.

5. **Option (E): 5, 4, 3, 2, 1**
   - The boss delivers the letters in the order 1, 2, 3, 4, 5.
   - The secretary types letter 5 first, which means letters 1, 2, 3, and 4 must still be in the pile.
   - Then, the secretary types letter 4, which means letters 1, 2, and 3 must still be in the pile.
   - Next, the secretary types letter 3, which means letters 1 and 2 must still be in the pile.
   - Then, the secretary types letter 2.
   - Finally, the secretary types letter 1.
   - This sequence is possible because the secretary can always type the top letter.
   - Therefore, option (E) is possible.

Conclusion:
The only sequence that is not possible is option (D).

The final answer is $\boxed{(D)}$