In the book I'm using on Catalan numbers, the author gives a scenario in order to develop the formula for Catalan numbers.
The scenario is that a boy has an empty jar. Every day he either puts in a dollar coin or takes on out for $2n$ days. At the end of the $2n$ days, the jar is empty. In how many ways can this happen?
I understand how he gets to the Catalan numbers. But there is one part where he says this:
note that if $n\gt 0$ then there had to be a first day other than the starting day when the jar was empty.
I'm having trouble understanding why this is the case. When I think of taking or putting a coin in a jar as the equality $y=x$, you could just add a coin in for the first $n$ days, then take one out for the remaining $2n - n = n$ days.
Why does there have to be a day other than the starting day in which the jar was $0$?