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(clippy::needless_range_loop)]
#![allow(unused_macros)]

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        #[allow(unused_mut)]
        let mut s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};

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

macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };

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

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

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

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

const MOD: usize = 998244353;
#[derive(Copy, Clone, Eq, PartialEq, std :: fmt :: Debug)]
struct ModInt(usize);
impl std::ops::Add for ModInt {
    type Output = Self;
    fn add(self, rhs: Self) -> Self {
        ModInt((self.0 + rhs.0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        self.0 += rhs.0;
        self.0 %= MOD;
    }
}
impl std::ops::Sub for ModInt {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self {
        ModInt((self.0 + MOD - rhs.0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        self.0 += MOD - rhs.0;
        self.0 %= MOD;
    }
}
impl std::ops::Mul for ModInt {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self {
        ModInt((self.0 * rhs.0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        self.0 *= rhs.0;
        self.0 %= MOD;
    }
}
impl std::ops::Div for ModInt {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        ModInt((self.0 * rhs.pow(MOD - 2).0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        self.0 *= rhs.pow(MOD - 2).0;
        self.0 %= MOD;
    }
}
impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.write_str(&self.0.to_string())
    }
}
impl ModInt {
    fn pow(&self, power: usize) -> ModInt {
        let mut acc_base = *self;
        let mut acc_pow = power;
        let mut res = ModInt(1);
        while acc_pow > 0 {
            if acc_pow & 1 == 1 {
                res *= acc_base;
            }
            acc_base *= acc_base;
            acc_pow >>= 1;
        }
        res
    }
}
struct ModIntFact {
    memo: Vec<ModInt>,
    memo_inv: Vec<ModInt>,
    size: usize,
}
impl ModIntFact {
    fn new(size: usize) -> Self {
        let mut memo = vec![ModInt(0); size + 1];
        memo[0] = ModInt(1);
        for n in 1..=size {
            memo[n] = memo[n - 1] * ModInt(n);
        }
        let memo_inv = memo.iter().map(|x| x.pow(MOD - 2)).collect();
        Self {
            memo,
            memo_inv,
            size,
        }
    }
    fn fact(&self, n: usize) -> ModInt {
        self.memo[n]
    }
    fn fact_inv(&self, n: usize) -> ModInt {
        self.memo_inv[n]
    }
    fn ncr(&self, n: usize, r: usize) -> ModInt {
        self.memo[n] * self.memo_inv[r] * self.memo_inv[n - r]
    }
}

fn main() {
    input! {
        n: usize,
        m: usize,
        k: usize,
    }
    let m = ModInt(m);
    let fact = ModIntFact::new(210000);
    let ans: ModInt = (0..=k).map(|k| m * fact.ncr(n - 1, k) * (m - ModInt(1)).pow(n - 1 - k)).fold(ModInt(0), |x, y| x + y);
    println!("{}", ans);
}