How many ways are there of choosing $k$ distinct items from a set of $n$? Specifically, say I have the integers $1,2,3,\dots,n$ (a set of $n$ integers). I want to select numbers one after another (not at the same time) until I have $k$ distinct numbers. How many ways are there of doing this? Someone told me $nPk$ but I don't understand why. Is there another way to approach it?

From $n$ distinct elements we can choose $k$-elements in $\binom{n}{k}$ ways and after permuting them in $k!$ ways we get that there exists 
$$\binom{n}{k}k!=\frac{n!}{k!(n-k)!}k!=\frac{n!}{(n-k)!}$$
possibilities
