Bingo for the Win!

Bingo is a game of chance for multiple players. Each player receives a sheet with some numbers, and a game master calls out these numbers in a random order. Players cross off the numbers they have heard, and the first player to cross off all their numbers wins the game. This basic version is fairly sedate. Players only need to stay awake.

In this problem we analyze a more dynamic version called Speed Bingo. The game master also calls out the numbers on the sheets in a random order. However, whenever a number is called, only the first player to signal that they have that number may cross it off their sheet. If a player has the same number multiple times, only one copy may be crossed off at a time. When several players have the same number, the player with the fastest reaction time has the advantage. But how big is that advantage? Finding out is the goal of this problem.

There are $n$ players, each receiving a sheet of $k$ numbers. The numbers may repeat, and different players may receive different sheets. Player $1$ reacts faster than player $2$, who reacts faster than player $3$, and so on, with player $n$ being the slowest. The following figure shows three players who each receive four numbers:

When the number $1$ is called for the first time, player $1$ crosses it off, since player $1$ is faster. When $1$ is called the second time, player $2$ crosses it off. On average, player $1$ is expected to do better than players $2$ and $3$, because both of them need numbers that player $1$ reaches first. Players $2$ and $3$ share no numbers, so their results are independent of each other, even though player $2$ is faster than player $3$.

Suppose the game continues until every player has crossed off all of their numbers, that is, until all $n \cdot k$ numbers on all sheets (counting repeats) have been read. If the order of the numbers is uniformly random, what is the probability that each player finishes last?

The input describes a single game of Speed Bingo. The first line contains two integers $n$ and $k$, the number of players and the number of numbers on each sheet ($1 \le n \le 100$, $1 \le k \le 1000$). The next $n$ lines each contain $k$ integers. The $i$th line gives the numbers on the sheet of the $i$th player. All numbers are between $1$ and $10^9$, inclusive.

Output $n$ lines, one for each player. The $i$th line contains the probability that player $i$ finishes last. All values must be accurate to an absolute error of at most $10^{-6}$.