Why do we use ${n\choose k}$ for a binomial distribution instead of ${n+k-1\choose k}$?

Question

I am trying to get my head around this. In my understanding a binomial distribution uses replacement and ${n\choose k}$ precisely states that there's no repetition and that's not the case with a coin toss for instance.

Thanks in advance.

Answer 1

In a binomial distribution, the choice of $k$ is the choice of which trials succeed. There can be no repetition because the same trial cannot succeed multiple times.

Answer 2

The probability of getting exactly $k$ successes in $n$ trials is given by the probability mass function:

$Pr(X=k)={n \choose k}p^{k}(1-p)^{n-k}$

i.e. of out $n$ choose ${{exactly}}$ $k$ to succeed with probability $p$.