Score : $500$ points

Problem Statement

We have a sequence of length $N$ consisting of integers between $0$ and $M$, inclusive: $A=(A_1,A_2,\dots,A_N)$.

Snuke will perform the following operations 1 and 2 in order.

1. For each $i$ such that $A_i=0$, independently choose a uniform random integer between $1$ and $M$, inclusive, and replace $A_i$ with that integer.
2. Sort $A$ in ascending order.

Print the expected value of $A_K$ after the two operations, modulo $998244353$.

Constraints

• $1\leq K \leq N \leq 2000$
• $1\leq M \leq 2000$
• $0\leq A_i \leq M$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$ $K$

$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the expected value of $A_K$ after the two operations, modulo $998244353$.

3 5 2
2 0 4

3

In the operation 1, Snuke will replace $A_2$ with an integer between $1$ and $5$. Let $x$ be this integer.

• If $x=1$ or $2$, we will have $A_2=2$ after the two operations.
• If $x=3$, we will have $A_2=3$ after the two operations.
• If $x=4$ or $5$, we will have $A_2=4$ after the two operations.

Thus, the expected value of $A_2$ is $\frac{2+2+3+4+4}{5}=3$.

2 3 1
0 0

221832080

The expected value is $\frac{14}{9}$.

10 20 7
6 5 0 2 0 0 0 15 0 0

617586310