To solve this problem, we need to consider the constraints: at least one D and at least one E must be included in the selection of 5 letters from the word DELIVERED.

The word DELIVERED consists of the letters: D, E, L, I, V, E, R, E, D.

We have 2 D's and 3 E's in the word.

We consider different scenarios based on the number of D's and E's selected:

1. 1 D and 1 E: Choose 3 more letters from the remaining 4 letters (L, I, V, R). This can be done in inom{4}{3} = 4 ways.
2. 1 D and 2 E's: Choose 2 more letters from the remaining 4 letters. This can be done in inom{4}{2} = 6 ways.
3. 1 D and 3 E's: Choose 1 more letter from the remaining 4 letters. This can be done in inom{4}{1} = 4 ways.
4. 2 D's and 1 E: Choose 2 more letters from the remaining 4 letters. This can be done in inom{4}{2} = 6 ways.
5. 2 D's and 2 E's: Choose 1 more letter from the remaining 4 letters. This can be done in inom{4}{1} = 4 ways.
6. 2 D's and 3 E's: No more letters needed. This is 1 way.

Adding these possibilities gives the total number of selections:

\(4 + 6 + 4 + 6 + 4 + 1 = 25.\)