A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct $3$-letter words can be made from a subset of the letters $c, o, m, b, o$, where each letter in the list is used no more than the number of times it appears?
To determine the number of distinct 3-letter words that can be made from the subset of letters \( \{c, o, m, b, o\} \), we need to consider different cases based on the number of times the letter 'o' is used. 

### Case 1: No 'o's are used
If no 'o's are used, we are left with the letters \( \{c, m, b\} \). We need to form a 3-letter word using these 3 letters. The number of permutations of 3 distinct letters is given by:
\[
3! = 6
\]

### Case 2: One 'o' is used
If one 'o' is used, we need to choose 2 more letters from the remaining set \( \{c, m, b\} \). The number of ways to choose 2 letters from 3 is given by the binomial coefficient:
\[
\binom{3}{2} = 3
\]
For each choice of 2 letters, we can form a 3-letter word by permuting the 3 letters (including the 'o'). The number of permutations of 3 letters is:
\[
3! = 6
\]
Thus, the total number of 3-letter words in this case is:
\[
\binom{3}{2} \cdot 3! = 3 \cdot 6 = 18
\]

### Case 3: Two 'o's are used
If two 'o's are used, we need to choose 1 more letter from the remaining set \( \{c, m, b\} \). The number of ways to choose 1 letter from 3 is given by the binomial coefficient:
\[
\binom{3}{1} = 3
\]
For each choice of 1 letter, we can form a 3-letter word by permuting the 3 letters (including the two 'o's). The number of permutations of 3 letters where 2 letters are identical is given by:
\[
\frac{3!}{2!} = 3
\]
Thus, the total number of 3-letter words in this case is:
\[
\binom{3}{1} \cdot \frac{3!}{2!} = 3 \cdot 3 = 9
\]

### Total Number of 3-Letter Words
Adding up the number of 3-letter words from all cases, we get:
\[
6 + 18 + 9 = 33
\]

Therefore, the total number of distinct 3-letter words that can be made is:
\[
\boxed{33}
\]