Form an 8 letter word using A,B,C,D,E, if the letters in the word must appear in alphabetical order Form an 8 letter word using A,B,C,D,E, where each letter can be used multiple times. How many words can I form if the letters in the word must appear in alphabetical order?
For example: AABBDDDE is acceptable, BBBACCCE is not acceptable.
The only way I can think of to count this is to draw a table with the number of occurrences of each letter, then calculate the permutations of the letter positions for each row.
Is there an easier way to solve this question?

Straightforward approach, where you almost don't need to think (Alex's and Alessandro Cigna's answers require some thinking).
Let $f(n,k)$ be the number of the desired strings with $n$ letters length with $k$ letters allowed. We have
$$f(1,k)=k,\quad f(n,k)=\sum\limits_{i=0}^{k-1}f(n-1,k-i).$$

Table for f(n,k)
n\k|      1 |      2 |      3 |      4 |      5
---+--------+--------+--------+--------+--------
 1 |      1 |      2 |      3 |      4 |      5
 2 |      1 |      3 |      6 |     10 |     15
 3 |      1 |      4 |     10 |     20 |     35
 4 |      1 |      5 |     15 |     35 |     70
 5 |      1 |      6 |     21 |     56 |    126
 6 |      1 |      7 |     28 |     84 |    210
 7 |      1 |      8 |     36 |    120 |    330
 8 |        |        |        |        |    495

