A9653 | Bingo!

The game of bingo is played on a $5×5$ square grid filled with distinct numbers between $1$ and $75$ . In this problem you will consider a generalized version played on an $n×n$ grid with distinct numbers between $1$ and $m$ $(m>=n^{2})$ .

A player begins by selecting a randomly generated bingo grid (generated uniformly among all available grids). Then $k$ distinct numbers between $1$ and $m$ will be called at random (called uniformly among all available sets of $k$ numbers). For each called number that appears on the grid, the player marks that cell. The score at the end is 2 raised to the power of (number of completely marked rows plus number of completely marked columns).

Determine the expected value of the score. The expected score may be very large. If the expected score is larger than $10^{99}$ , print $10^{99}$ instead (for example as "1e99" without the quotes).

Input will consist of three integers $n$ , $m$ , $k$ $(1<=n<=300; n^{2}<=m<=100000; n<=k<=m)$ .

Print the smaller of $10^{99}$ and the expected score. Your answer must be correct within an absolute or relative error of $10^{-9}$ .