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?
 A: Observe that the type of word that you want is univocally decided by the numbers of letters A,B,C,D and E. Then the problem is the same of ask in how many ways can you write 8 as sum of 5 numbers, and the answer is ${12\choose 8}$, do you know why?
A: 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

A: The answer is the same if you count alphabetical words of length $13$ in which each letter must appear at least once (by adding/removing one copy of each letter).
To count these, imagine a list of $13$ slots (aka "stars") which will hold the letters. To specify a word, you only need to pick $4$ gaps from the $12$ interior gaps between slots to specify the $4$ places (aka "bars") where the letter changes in the word (i.e. A to B, B to C, etc).
This can be done in $C(12,4) = 495$ ways.
A: You have $8+5-1=12$ slots, from which you need to choose $4$. Wach such choice determines the number of times each letter is repeated. For example, if you choose slots 1 to 4, you get all Es. Can you handle from here?
