Can't tell lists from sets in counting problems 
You throw five identical six-sided dice and write down the values showing, in nondecreasing
  order from left to right. For example, $22245$ means you rolled three $2$s, one $4$, and one $5$. How many outcomes are possible? How many in which all the values
  are different?

My first instinct is to say that  there are $6^5$ such words with repetition and $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot$ words without repetition. But it turns out that these "words" are actually sets so the solutions are $6 \text { multichoose } 5$ and $6 \text { choose } 5$, respectively. 
What terms in the statement of the problem point to the fact that we are counting sets, not words?  
 A: The values are put into ascending order from left to right, so 22345 is the same outcome as 22435, since 22435 will be rearranged into 22345. Therefore, the order of the numbers doesn't matter, just like in a set.
A: This is a classic stars and bars problem. If you have ten spaces and fill five of them with bars, then the others - stars - represent the outcomes of the throws. Reading from left to right the value starts at $1$, every time you pass a bar the value goes up by $1$.
Eg $*|*|*|*|*|$ translates to $12345$ while $||***|||**$  represents $33366$. The number of possible outcomes is then $\binom {10}5$. You might want to prove more fully that the translation between interpretations works both ways and doesn't leave anything out.
What matters in your problem, as is clear from the arrangement of outcomes into a standard form from which duplicates can readily be identified, is the number of twos which are thrown, not whether the two twos you get come first and third or fourth and fifth. The "words" retain the order, and count duplicates more than once.
