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)

#[allow(dead_code)]
fn is_whitespace(c: &char) -> bool {
    c.is_whitespace()
}

#[allow(dead_code)]
fn is_eol(c: &char) -> bool {
    ['\r', '\n'].contains(c)
}

#[allow(unused_macros)]
macro_rules! input {
    ($var:ident, $($r:tt)*) => {
        input_inner!{$var, $($r)*}
    };

    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        #[allow(unused_mut, unused_variables, bare_trait_objects)]
        let mut next = move |f: &Fn(&char)->bool| -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|f(&c))
                .take_while(|c|!f(&c))
                .collect()
        };
        input_inner!{next, $($r)*};
    };
}

#[allow(unused_macros)]
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

#[allow(unused_macros)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; eol ]) => {
        $next(&is_eol).split(' ').map(|x| x.parse::<$t>().unwrap()).collect::<Vec<_>>()
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, parser) => {
        $next
    };

    ($next:expr, $t:ty) => {
        $next(&is_whitespace).parse::<$t>().expect("Parse error")
    };
}

#[allow(unused_imports)]
use std::cmp::{max, min};
#[allow(unused_imports)]
use std::collections::{BinaryHeap, HashMap, HashSet};

#[allow(dead_code)]
struct Edge {
    from: usize,
    to: usize,
}

#[allow(dead_code)]
type Graph = Vec<Vec<usize>>;
#[allow(dead_code)]
const M: usize = 998244353;

struct Combination {
    modp: usize,
    mfact: Vec<usize>,
    rfact: Vec<usize>,
}

impl Combination {
    fn new(size: usize, modp: usize) -> Self {
        let mut rfact = vec![0; size + 1];
        let mut mfact = vec![0; size + 1];
        mfact[0] = 1;
        for i in 1..mfact.len() {
            mfact[i] = (mfact[i - 1] * i) % modp;
        }
        rfact[size] = Self::inv(mfact[size], modp);
        for i in (0..size).rev() {
            rfact[i] = rfact[i + 1] * (i + 1) % modp;
        }

        Combination {
            modp: modp,
            mfact: mfact,
            rfact: rfact,
        }
    }

    fn pow(mut x: usize, mut n: usize, modp: usize) -> usize {
        let mut ret = 1;
        while n > 0 {
            if n & 1 > 0 {
                ret *= x;
                ret %= modp;
            }
            x *= x;
            x %= modp;
            n >>= 1;
        }
        ret
    }

    fn inv(x: usize, modp: usize) -> usize {
        Self::pow(x, modp - 2, modp)
    }

    fn C(&self, p: usize, q: usize) -> usize {
        if p < q {
            return 0;
        }
        (((self.mfact[p] * self.rfact[q]) % self.modp) * self.rfact[p - q]) % self.modp
    }
}

fn main() {
    input! {
        n: usize, m: usize, k: usize,
    }
    let comb = Combination::new(200001, M);
    let mut sum = 0;
    let mut mp = vec![0; n];
    mp[0] = 1;
    for i in 1..n {
        mp[i] = mp[i - 1] * (m - 1);
        mp[i] %= M;
    }
    for i in 0..k + 1 {
        let mut s = m * mp[n - i - 1];
        s %= M;
        s *= comb.C(n - 1, i);
        s %= M;
        sum += s;
        sum %= M;
    }
    println!("{}", sum);
}