Problem:
Score : 500 points

Problem Statement

There are N blocks arranged in a row. Let us paint these blocks.

We will consider two ways to paint the blocks different if and only if there is a block painted in different colors in those two ways.

Find the number of ways to paint the blocks under the following conditions:

For each block, use one of the M colors, Color 1 through Color M, to paint it. It is not mandatory to use all the colors.

There may be at most K pairs of adjacent blocks that are painted in the same color.

Since the count may be enormous, print it modulo 998244353.

Constraints

All values in input are integers.

1 \leq N, M \leq 2 \times 10^5

0 \leq K \leq N - 1

Input

Input is given from Standard Input in the following format:

N M K

Output

Print the answer.

Sample Input 1

3 2 1

Sample Output 1

6

The following ways to paint the blocks satisfy the conditions: 112, 121, 122, 211, 212, and 221. Here, digits represent the colors of the blocks.

Sample Input 2

100 100 0

Sample Output 2

73074801

Sample Input 3

60522 114575 7559

Sample Output 3

479519525

Solution in Rust (1.42.0) (judge verdict: Accepted)

use proconio::{input, fastout};

const MAX: usize = 200_000;
const MOD: usize = 998244353;

struct ModComb {
    max: usize,
    modulus: usize,
    fact: Vec<usize>,
    finv: Vec<usize>,
    inv: Vec<usize>,
}

impl ModComb {
    fn new(max: usize, modulus: usize) -> Self {
        let mut mc = Self {
            max,
            modulus,
            fact: vec![1; max],
            finv: vec![1; max],
            inv: vec![0; max],
        };
        mc.init_table();
        mc
    }

    fn init_table(&mut self) {
        self.inv[1] = 1;
        for i in 2..self.max {
            self.fact[i] = self.fact[i-1] * i % self.modulus;
            self.inv[i] = self.modulus - self.inv[self.modulus%i] * (self.modulus / i) % self.modulus;
            self.finv[i] = self.finv[i-1] * self.inv[i] % self.modulus;
        }
    }

    fn comb(&self, n: usize, k: usize) -> usize {
        if n < k {
            0
         } else {
            self.fact[n] * (self.finv[k] * self.finv[n-k] % self.modulus) % self.modulus
        }
    }
}

fn mod_pow(a: usize, n: usize, m: usize) -> usize {
    if n == 0 {
        1
    } else if n % 2 == 0 {
        let x = mod_pow(a, n / 2, m);
        x * x % m
    } else {
        let x = mod_pow(a, n / 2, m);
        a * ((x * x) % m) % m
    }
}

#[fastout]
fn main() {
    input! {
        n: usize,
        m: usize,
        k: usize,
    }

    let mut ans = 0usize;
    let mc = ModComb::new(MAX, MOD);
    for i in 0..=k {
        let mut num = m;
        num = num * mc.comb(n-1, i) % MOD;
        num = num * mod_pow(m-1, n-1-i, MOD) % MOD;
        ans = (ans + num) % MOD;
    }

    println!("{}", ans);

}