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)

pub struct ModUtil {
    fact: Vec<usize>,
    inv_fact: Vec<usize>,
    module: usize,
}

#[allow(dead_code)]
impl ModUtil {
    pub fn new(size: usize, module: usize) -> ModUtil {
        let mut fact = vec![0; size + 1];
        let mut inv_fact = vec![0; size + 1];
        fact[0] = 1;
        for i in 1..size + 1 {
            fact[i] = i * fact[i - 1] % module;
        }
        inv_fact[0] = 1;
        for i in 1..size + 1 {
            inv_fact[i] = ModUtil::mod_pow(fact[i], module - 2, module);
        }
        ModUtil {
            fact: fact,
            inv_fact: inv_fact,
            module: module,
        }
    }

    pub fn permutation(&self, n: usize, k: usize) -> usize {
        assert!(n >= k);
        self.fact[n] % self.module * self.inv_fact(n - k) % self.module
    }
    pub fn combination(&self, n: usize, k: usize) -> usize {
        assert!(n >= k);
        self.fact[n] * self.inv_fact[k] % self.module * self.inv_fact[n - k] % self.module
    }

    pub fn fact(&self, x: usize) -> usize {
        self.fact[x]
    }
    pub fn inv(&self, x: usize) -> usize {
        ModUtil::mod_pow(x, self.module - 2, self.module)
    }
    pub fn inv_fact(&self, x: usize) -> usize {
        self.inv_fact[x]
    }

    pub fn mod_pow(x: usize, n: usize, module: usize) -> usize {
        if n == 0 {
            return 1;
        }
        let mut res: usize = ModUtil::mod_pow(x, n >> 1, module) % module;
        res = (res * res) % module;
        if n & 1 == 1 {
            res *= x;
        }
        res % module
    }
}

const MOD: usize = 998244353;
const MAX_NM: usize = 200_010;
#[derive(Default)]
//NOTE
//declare variables to reduce the number of parameters for dp and dfs etc.
struct Solver {}

impl Solver {
    fn solve(&mut self) {
        let stdin = std::io::stdin();
        let mut scn = Scanner {
            stdin: stdin.lock(),
        };
        let n: usize = scn.read();
        let m: usize = scn.read();
        let k: usize = scn.read();
        let mut mod_util = ModUtil::new(MAX_NM, MOD);
        let mut result = 0;
        for x in 0..=k {
            let perm = m % MOD * mod_util.fact(n - 1) %MOD * mod_util.inv_fact(x) % MOD * mod_util.inv_fact(n - 1 - x) % MOD;
            let cnt = ModUtil::mod_pow(m - 1, n - 1 - x, MOD) % MOD;
            result += perm * cnt % MOD;
            result %= MOD;
            
        }
        println!("{}", result % MOD);
    }
}

fn main() {
    std::thread::Builder::new()
        .stack_size(64 * 1024 * 1024) // 64MB
        .spawn(|| {
            let mut solver = Solver::default();
            solver.solve()
        })
        .unwrap()
        .join()
        .unwrap();
}
pub mod utils {
    const DYX: [(isize, isize); 8] = [
        (0, 1),   //right
        (1, 0),   //down
        (0, -1),  //left
        (-1, 0),  //top
        (1, 1),   //down right
        (-1, 1),  //top right
        (1, -1),  //down left
        (-1, -1), //top left
    ];

    pub fn try_adj(y: usize, x: usize, dir: usize, h: usize, w: usize) -> Option<(usize, usize)> {
        let ny = y as isize + DYX[dir].0;
        let nx = x as isize + DYX[dir].1;
        if ny >= 0 && nx >= 0 {
            let ny = ny as usize;
            let nx = nx as usize;
            if ny < h && nx < w {
                Some((ny, nx))
            } else {
                None
            }
        } else {
            None
        }
    }
}

//snippet from kenkoooo
pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn read_line(&mut self) -> String {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b'\n' || b == b'\r')
            .take_while(|&b| b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }

    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}