Binomial Series and the nth derivative

Question

When differentiating a binomial series of the form $(1+x)^k$, why is the the $n$th derivative equal to $k(k-1) ... (k-n+1)$? I don't understand where there is $+1$ in the end of $(k-n+1)$ is coming from.

Answer 1

The product contains $n$ terms since you've differentiated $n$ times. But the numbers subtracted from $k$ in each factor begins at $0,$ not $1.$ Hence the last factor is $$k-(n-1)=k-n+1.$$