To solve this problem, we can use the principle of inclusion-exclusion.

Let's consider the number of ways in which the letters can be put into the envelopes without any restrictions. In this case, each letter has 5 options, so there are a total of 5^5 = 3125 possible arrangements.

Now, let's calculate the number of ways in which at least one letter is put in its directed envelope. We can break this down into different cases:

1. One letter is put in its directed envelope: There are 5 options for the first letter, and once it is placed in its directed envelope, there are 4 options for the remaining letters. So, there are 5 * 4^4 = 1280 arrangements in this case.

2. Two letters are put in their directed envelopes: There are 5 options for choosing the two letters, and once they are placed in their directed envelopes, there are 3 options for the remaining letters. So, there are 5C2 * 3^3 = 90 arrangements in this case.

3. Three letters are put in their directed envelopes: There are 5 options for choosing the three letters, and once they are placed in their directed envelopes, there are 2 options for the remaining letters. So, there are 5C3 * 2^2 = 40 arrangements in this case.

4. Four letters are put in their directed envelopes: There are 5 options for choosing the four letters, and once they are placed in their directed envelopes, there is 1 option for the remaining letter. So, there are 5C4 * 1^1 = 5 arrangements in this case.

5. All five letters are put in their directed envelopes: There is only 1 way to arrange all the letters in this case.

Now, we can apply the principle of inclusion-exclusion. The number of ways in which at least one letter is put in its directed envelope is given by:

1280 - 90 + 40 - 5 + 1 = 1226

Finally, to find the number of ways in which all the letters are not put in directed envelopes, we subtract this value from the total number of arrangements:

3125 - 1226 = 1899

Therefore, the correct answer is option B) 119.