I have a question about the binomial coefficient solution to the generalization of the egg dropping problem (n eggs, k floors)
So there are two cases, if we drop an egg. Either it breaks, so we have $x-1$ trials left and $n-1$ eggs, or it does not break, so we have $x-1$ trials left and $n$ eggs.
Thus we have the following recursive expression: $$ f(x,n)=1+f(x-1,n)+f(x-1,n-1) $$
I cannot understand why we are adding the terms $f(x-1,n)$ and $f(x-1,n-1)$. As only one of the two contingencies happens, (the first egg breaks or not), why do we use simultaneously both terms in the sum.
Can someone please explain that in detail?
Thank you in advance