In today's quiz (about combinatorics) there was a question I was not able to solve mathematically:
Find the number of words with length 3 over an alphabet $ \Sigma = \{1,2,...,10\} $ where $x \lt y \le z$. ($x$ is the first letter, and so on).
How can I solve this in an efficient way? Of course you can just count it like:
$9 \times 10 \times 10 \rightarrow$ 1 possiblity
$8 \times \{9,10\} \times \{9,10,10\} \rightarrow$ 2 + 1 possibilities
....
But how do I solve it with "combinatoric" methods?