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How is it possible to arrange $0$ objects? Well, by doing nothing. Doing nothing *is something you can do*, and you can do it in exactly one way.

Another way to see it is to consider that it is also the number of ways you can *rearrange* the objects in a row. One way to rearrange the objects is, again, doing nothing. If there is only one object, doing nothing is indeed the only way to rearrange the objects, therefore $1!=1$. And if there are no objects at all, again, doing nothing is your only option, thus also $0!=1$.

This is different to the situation when dividing by zero. If you distribute $m$ cakes to $n$ children so that no cake is left, each child gets $m/n$ cakes. Now what happens if $n=0$? Well, if $m$ is nonzero, it is simply not possible: If you have no children to distribute your cake to, you simply won't get rid of your cake, as there's no one there to take it. On the other hand, if $m=0$ as well (you have no cake) you *can* distribute your zero cakes to the zero children; however you can't tell how many cakes each child will get, as any number of cakes per child will work.

Also, generally $n!$ is not *defined* to be the number of arrangements, but simply as the product $1\cdot 2\cdot 3 \cdots n$. It just turns out that this is exactly the number of arrangements of $n$ objects in a row.

Now with that definition, how do we see that $0!=1$? Well, clearly $0!$ is the empty product, which by definition is $1$.

Alternatively you can observe that obviously for $n>0$, $(n+1)! = (n+1)n!$. Therefore the obvious definition of $0!$ is so that this relation still holds, that is $1! = 1\cdot 0!$. Given that $1!=1$, it immediately follows that $0!=1$.