Binomial Coefficient and the Number of Combinations when $k=0$

Question

This must have been asked before, but I couldn't find it. I am looking for an explanation, intuitive if possible (less technical), for why $\binom{n}{0}$ equals $1$. I understand that mathematically $0!=1$, but I am looking for the verbal explanation. $\binom{n}{k}$ is the number of combinations of size $k$ from a set of $n$ objects, what is the meaning of choosing $0$ objects out of $n$? Why is it $1$?

Answer 1

Another definiton of the binomial coefficient is: How many subsets with $k$ elements does a Set with $n$ elements have? Well, there is only one set with zero elements, which is the empty set and the empty set is a subset of every set. Thus, the empty set qualifies as subset for every set with arbitrary size $n$ and as long as $k$ is $0$, there will be only one solution.

Answer 2

Just think of sitting in front of $n$ distinct objects. In how many ways can you pick up one? Well, ${n \choose 1}=n$ ways, since you could either pick up the first, or the second, etc. The same reasoning of course goes for $k$ objects. Now imagine NOT picking up anything. In how many WAYS can you do this, i.e do nothing? Well in one way, just don't pick anything up.

If you want to think about unordered strings without repititions, or something like that, you could think about how many strings of zero charachters there are. That would be the empty string, and there is only one.

For a more set theoretical explaination, I believe Brians answer, in the comments above, is a good one, namely that a set has only one subset with zero members.

Edit: That there is only one unique empty set can be proven. Even though there might not be a nice proof that there's only one way to do nothing, the reasoning is the same. Think about how there could be more than one way to do nothing, what would make those ways different? (A possible 'proof' of this would most likely reduce to proving that there is only one empty set.)

I hope this was verbal enough. I am not sure how to explain it informally in a better way.