Bingo!

题意翻译

有一个$n×n(n≤300)$的棋盘和$1~m(n^2≤m≤100000)$ 这些数字。棋盘首先会被随机生成,即从“填着值域$1 \sim m$的数字且$n^2$个数字两两不同”的所有方案中随机选一个。 然后你会从$1\sim m$中随机选出$k(n≤k≤m)$个不同的数,并且报出来。报出一个数字,如果在棋盘中出现过,就把对应格涂黑。设$t$为完全染黑的行数、完全染黑的列数的和,你的最终得分为$2^t$求得分的期望。如果超过$10^{99}$,输出$1e99$

题目描述

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).

输入输出格式

输入格式

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

输出格式

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

输入输出样例

输入样例 #1

1 2 1

输出样例 #1

2.5

输入样例 #2

2 4 3

输出样例 #2

4

输入样例 #3

7 59164 40872

输出样例 #3

3.1415926538