Suppose I have $N$ numbered balls (i.e., one ball is labelled "1", one ball is labelled "2", etc., like Bingo balls). I am trying to identify the probability distribution of pulling $k$ of the balls on the draw that corresponds to the number on the ball, i.e., "Ball 1" on "Draw 1", "Ball 2" on "Draw 2", etc., without replacement.
The case where $N=2$ is pretty trivial: there is a 0.5 chance of pulling all balls in order (1,2) and a 0.5 chance of pulling no balls in order (2,1). There is a 0 chance of pulling one ball in order.
If we move up to $N=3$, then there is a $1/3!$ chance of pulling all three balls in order (1,2,3), a 0 chance of pulling two of the balls at the right time, a 0.5 chance of getting one of the balls (1,3,2 gets the first ball on Draw 1; or 2,1,3 gets the third ball on Draw 3; or 3,2,1 gets the second ball on Draw 2), and a $1/3$ chance of getting none.
Some patterns are clear, there is a $1/N!$ chance of getting all balls on the right-numbered Draw, and a 0 chance of getting N-1 balls on the appropriate Draws. But can we generalize to calculate the probability distribution for larger numbers of balls/draws?
