Nezzar and Chocolate Bars

题意翻译

给定 $n$ 个长度为 $a_i$ 的巧克力，每次以正比于 $a_i$ 的概率取得一个巧克力，然后在 $(0,a_i)$ 中随机选择一个实数 $r$ 并将其分成 $r,a_i-r$ 两个部分放回。 计算使得所有巧克力的长度均小于 $k$ 的期望操作次数。 $n\le 50,\sum a_i,k\le 2000$

题目描述

Nezzar buys his favorite snack — $ n $ chocolate bars with lengths $ l_1,l_2,\ldots,l_n $ . However, chocolate bars might be too long to store them properly! In order to solve this problem, Nezzar designs an interesting process to divide them into small pieces. Firstly, Nezzar puts all his chocolate bars into a black box. Then, he will perform the following operation repeatedly until the maximum length over all chocolate bars does not exceed $ k $ . - Nezzar picks a chocolate bar from the box with probability proportional to its length $ x $ . - After step $ 1 $ , Nezzar uniformly picks a real number $ r \in (0,x) $ and divides the chosen chocolate bar into two chocolate bars with lengths $ r $ and $ x-r $ . - Lastly, he puts those two new chocolate bars into the black box. Nezzar now wonders, what is the expected number of operations he will perform to divide his chocolate bars into small pieces. It can be shown that the answer can be represented as $ \frac{P}{Q} $ , where $ P $ and $ Q $ are coprime integers and $ Q \not \equiv 0 $ ( $ \bmod 998\,244\,353 $ ). Print the value of $ P\cdot Q^{-1} \mod 998\,244\,353 $ .

输入输出格式

输入格式

The first line contains two integers $ n $ and $ k $ ( $ 1 \le n \le 50, 1 \le k \le 2000 $ ). The second line contains $ n $ integers $ l_1, l_2, \ldots, l_n $ ( $ 1 \le l_i $ , $ \sum_{i=1}^{n} l_i \le 2000 $ ).

输出格式

Print a single integer — the expected number of operations Nezzar will perform to divide his chocolate bars into small pieces modulo $ 998\,244\,353 $ .

输入输出样例

输入样例 #1

1 1
2

输出样例 #1

4

输入样例 #2

1 1
1

输出样例 #2

0

输入样例 #3

1 5
1234

输出样例 #3

15630811

输入样例 #4

2 1
2 3

输出样例 #4

476014684

输入样例 #5

10 33
10 20 30 40 50 60 70 80 90 100

输出样例 #5

675105648

题意翻译

给定 $n$ 个长度为 $a_i$ 的巧克力，每次以正比于 $a_i$ 的概率取得一个巧克力，然后在 $(0,a_i)$ 中随机选择一个实数 $r$ 并将其分成 $r,a_i-r$ 两个部分放回。 计算使得所有巧克力的长度均小于 $k$ 的期望操作次数。 $n\le 50,\sum a_i,k\le 2000$