Nezzar and Chocolate Bars
Description
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 $P\cdot Q^{-1} \mod 998\,244\,353$.
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 $l_1, l_2, \ldots, l_n$ ($1 \le l_i$, $\sum_{i=1}^{n} l_i \le 2000$).
Print a single integer — the expected number of operations Nezzar will perform to divide his chocolate bars into small pieces modulo $998\,244\,353$.
Input
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 $l_1, l_2, \ldots, l_n$ ($1 \le l_i$, $\sum_{i=1}^{n} l_i \le 2000$).
Output
Print a single integer — the expected number of operations Nezzar will perform to divide his chocolate bars into small pieces modulo $998\,244\,353$.
Samples
1 1
2
4
1 1
1
0
1 5
1234
15630811
2 1
2 3
476014684
10 33
10 20 30 40 50 60 70 80 90 100
675105648