TITLE: Picking zero out of zero elements
QUESTION [23 upvotes]: How can this expression$$\binom{0}{0}=1$$
be logical? 
I do not know if this is good example:

Probability of selecting one blue candy from a jar, which contains 5 green candies.
$$\frac{\binom{0}{0}}{\binom{5}{1}}=\frac{1}{5}$$

REPLY [2 votes]: Definition 1.1 of https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch1.pdf is helpful to understand this.
Consider the power set $\mathcal{P}(X)$ of a set $X$, which is the set of all subsets of $X$. Note that this includes the empty set $\emptyset$. Picking any number (including 0) of items from $X$ is equivalent to picking one of the items in $\mathcal{P}(X)$. Thus, the number of ways of picking any number $n$ of items from $X$ is equal to the number of occurrences in $\mathcal{P}(X)$ of a set of length $n$.
If we select $n=0$, there is only one set of length $0$, namely the empty set $\emptyset$ which exists only once in $\mathcal{P}(X)$. Thus follows ${m \choose 0}=1$ for any $m \in \mathbb{N}$.