I am trying to understand the model answer for the following question
Given some alphabet $Σ$ and $w ∈ Σ^∗$, the number of string $x ∈ Σ^∗$ are there such that $|x| ≤ |w|$
Solution: $\sum_{i=0}^{|w|} |Σ|^i $
lets assume $|Σ| = 10$ and $|w| = 2$. Does this mean $\sum_{i=0}^{2} (|10|^0 + |10|^1 +|10|^2) $ or does this mean $\sum_{i=0}^{2} (|1|^0 + |2|^1 +|3|^2)$ (the size of the language at that index)
furthermore what is $x$ and $w$ is it a string? a set of strings?
Both $x$ and $w$ are defined as $\in \Sigma^*$. Thus they are elements of the set of all strings over the alphabet $\Sigma$, which means they are strings.
Thus $|w|$ is the length of the string, not the cardinality of a set.
As @rici has already stated in the comment, the solution means in your example case $\sum_{i=0}^2 (10^i)$ which is $10^0 + 10^1 + 10^2$ (no cardinality for the number 10).
You have $10^0=1$ strings of length zero, the unique empty string.
You have $10^1=10$ strings of length one, those consisting of one of the possible letters.
You have $10^2=100$ strings of length two, because you have ten options for the first position and ten for the second.