Score : $500$ points

Problem Statement

There are $K$ boxes numbered $1$ to $K$. Initially, all boxes are empty.

Snuke has some balls with integers from $1$ to $N$ written on them. Among them, $a_i$ balls have the number $i$. Balls with the same integer written on them cannot be distinguished.

Snuke has decided to put all of his balls in the boxes. He wants the balls with the number $1$ to have a majority in every box. In other words, in every box, the number of balls with $1$ should be greater than that of the other balls.

Find the number of such ways to put the balls in the boxes, modulo $998244353$.

Two ways are distinguished when there is a pair of integers $(i,j)$ satisfying $1 \leq i \leq K, 1 \leq j \leq N$ such that the numbers of balls with $j$ in Box $i$ are different.

Constraints

• All values in input are integers.
• $1 \leq N \leq 10^5$
• $1 \leq K \leq 200$
• $1 \leq a_i < 998244353$

Input

Input is given from Standard Input in the following format:

$N$ $K$

$a_1$ $\cdots$ $a_N$

Output

Print the number of ways, modulo $998244353$, to put the balls in the boxes so that the balls with the number $1$ have a majority in every box.

2 2
3 1

2

• There are two ways to put the balls so that the balls with the number $1$ will be in the majority.
• One way to do so is to put two of the balls with $1$ in Box $1$ and the other one in Box $2$, and put the one ball with $2$ in Box $1$.

2 1
1 100

0

• There may be no way to put the balls in the boxes to meet the requirement.

20 100
1073813 90585 41323 52293 62633 28788 1925 56222 54989 2772 36456 64841 26551 92115 63191 3603 82120 94450 71667 9325

313918676

• Be sure to find the count modulo $998244353$.