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 std::io::Read;

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();

    let answer = solve(&buf);

    println!("{}", answer);
}

fn solve(input: &str) -> String {
    let mut iterator = input.split_whitespace();

    let n: usize = iterator.next().unwrap().parse().unwrap();
    let m: usize = iterator.next().unwrap().parse().unwrap();
    let k: usize = iterator.next().unwrap().parse().unwrap();

    let mc: ModComb = ModComb::init(n);

    let mut ans: usize = 0;
    let mut col: usize = m;
    for x in (0..n).rev() {
        let mut now = col;
        now = now * mc.calc(n - 1, x) % MOD;
        if x <= k {
            ans = (ans + now) % MOD;
        }
        col = col * (m - 1) % MOD;
    }

    return ans.to_string();
}

const MOD: usize = 998244353;

struct ModComb {
    fac: Vec<usize>,
    finv: Vec<usize>,
    inv: Vec<usize>,
}

impl ModComb {
    fn init(max: usize) -> ModComb {
        let mut fac = vec![1; max];
        let mut finv = vec![1; max];
        let mut inv = vec![1; max];
        for i in 2..max {
            fac[i] = fac[i - 1] * i % MOD;
            inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
            finv[i] = finv[i - 1] * inv[i] % MOD;
        }
        return ModComb {
            fac: fac,
            finv: finv,
            inv: inv,
        }
    }

    fn calc(&self, n: usize, k: usize) -> usize {
        if n < k {
            return 0;
        }
        return self.fac[n] * (self.finv[k] * self.finv[n - k] % MOD) % MOD;
    }
}