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Ex - Rearranging Problem

Score : $600$ points

Problem Statement

There are $N$ people called Person $1$, Person $2$, $\dots$, Person $N$, lined up in a row in the order of $(1,2,\dots,N)$ from the front. Person $i$ is wearing Color $c_i$.
Takahashi repeated the following operation $K$ times: choose two People $i$ and $j$ arbitrarily and swap the positions of Person $i$ and Person $j$.
After the $K$ operations have ended, the color that the $i$-th person from the front is wearing coincided with $c_i$, for every integer $i$ such that $1 \leq i \leq N$.
How many possible permutations of people after the $K$ operations are there? Find the count modulo $998244353$.

Constraints

• $2 \leq N \leq 200000$
• $1 \leq K \leq 10^9$
• $1 \leq c_i \leq N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$c_1$ $c_2$ $\dots$ $c_N$

Output

Print the answer.

4 1
1 1 2 1

3

Here is a comprehensive list of possible Takahashi's operations and permutations of people after each operation.

• Swap the positions of Person $1$ and Person $2$, resulting in a permutation $(2, 1, 3, 4)$.
• Swap the positions of Person $1$ and Person $4$, resulting in a permutation $(4, 2, 3, 1)$.
• Swap the positions of Person $2$ and Person $4$, resulting in a permutation $(1, 4, 3, 2)$.

3 3
1 1 2

1

Here is an example of a possible sequence of Takahashi's operations.

• In the $1$-st operation, he swaps the positions of Person $1$ and Person $3$, resulting in a permutation $(3, 2, 1)$.
  In the $2$-nd operation, he swaps the positions of Person $2$ and Person $3$, resulting in a permutation $(2, 3, 1)$.
  In the $3$-rd operation, he swaps the positions of Person $1$ and Person $3$, resulting in a permutation $(2, 1, 3)$.

Note that, during the sequence of operations, the color that the $i$-th person from the front is wearing does not necessarily coincide with $c_i$.

10 4
2 7 1 8 2 8 1 8 2 8

132