CF457D.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 1099, print 1099 instead (for example as "1e99" without the quotes).

宾果游戏（Bingo）在一个 $5 \times 5$ 的方格上进行，方格中填入 $1$ 到 $75$ 之间互不相同的数字。在本题中，你将考虑其推广版本：在一个 $n \times n$ 的方格上进行，方格中填入 $1$ 到 $m$（其中 $m \geq n^2$）之间互不相同的数字。

玩家首先随机选取一个宾果方格（所有可能的方格等概率生成）。随后，从 $1$ 到 $m$ 中随机选出 $k$ 个互不相同的数（所有大小为 $k$ 的子集等概率被选中）。对每个被叫出的、且出现在方格中的数字，玩家标记对应格子。最终得分为：$2$ 的幂次，其指数为被完全标记的行数与被完全标记的列数之和。

请计算该得分的期望值。期望得分可能非常大；若期望得分大于 $10^{99}$，则输出 1e99（不含引号）。

Input will consist of three integers n, m, k (1 ≤ n ≤ 300; _n_2 ≤ m ≤ 100000; n ≤ k ≤ m).

输入包含三个整数 $n$、$m$、$k$（$1 ≤ n ≤ 300$；$n^2 ≤ m ≤ 100000$；$n ≤ k ≤ m$）。

Print the smaller of 1099 and the expected score. Your answer must be correct within an absolute or relative error of 10 - 9.

输出 1099 与期望得分中的较小值。你的答案必须在绝对或相对误差 $10^{-9}$ 范围内正确。