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::{fastout, input};

#[fastout]
fn main() {
    type Mint = ModInt<M998244353>;
    input! {
        n: Mint, m: Mint, k: usize,
    }
    let mut f = fact::Fact::new();
    let mut ans = Mint::from(0);
    for i in 0..=k {
        let cnt = m * num::pow(m - 1, usize::from(n) - 1 - i);
        let nn = n - i;
        let rr = i.into();
        let nr = nn + rr - 1;
        let comb = f.fact(nr) / f.fact(rr) / f.fact(nr - rr);
        ans += cnt * comb;
    }
    println!("{}", ans);
}

mod fact {
    use num::traits::One;
    pub struct Fact<T: One>(Vec<T>);
    impl<T: One + Copy + From<usize> + Into<usize>> Fact<T> {
        pub fn new() -> Self {
            Self(vec![T::one()])
        }
        pub fn fact(&mut self, x: T) -> T {
            let a = self.0.len();
            let b: usize = x.into();
            for i in a..=b {
                self.0.push(self.0[i - 1] * T::from(i));
            }
            self.0[b]
        }
    }
}

use mod_int::*;
mod mod_int {
    use std::marker::PhantomData;
    #[macro_export]
    macro_rules! impl_modulus {
        ($t :ident ,$m :expr ) => {
            #[allow(dead_code)]
            pub struct $t;
            impl Modulus for $t {
                const VALUE: u64 = $m;
            }
        };
    }
    pub trait Modulus {
        const VALUE: u64;
    }
    impl_modulus!(M1000000007, 1_000_000_007);
    impl_modulus!(M998244353, 998_244_353);
    pub struct ModInt<M: Modulus>(u64, PhantomData<M>);
    impl<M: Modulus> std::marker::Copy for ModInt<M> {}
    impl<M: Modulus> std::clone::Clone for ModInt<M> {
        fn clone(&self) -> Self {
            Self(self.0, PhantomData::<M>)
        }
    }
    macro_rules !impl_primitive {($($t :ty ) *) =>($(impl <M :Modulus >From <$t >for ModInt <M >{fn from (x :$t ) ->Self {Self ((x as $t ) .rem_euclid (M ::VALUE as $t ) as u64 ,PhantomData ::<M >,) } } impl <M :Modulus >From <ModInt <M >>for $t {fn from (x :ModInt <M >) ->$t {x .0 as $t } } ) *) }
    impl_primitive !(usize u8 u16 u32 u64 isize i8 i16 i32 i64 ) ;
    impl<M: Modulus> std::str::FromStr for ModInt<M> {
        type Err = std::num::ParseIntError;
        fn from_str(s: &str) -> Result<Self, Self::Err> {
            Ok(Self::from(s.parse::<u64>()?))
        }
    }
    impl<M: Modulus> std::fmt::Debug for ModInt<M> {
        fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
            write!(f, "{}", self.0)
        }
    }
    impl<M: Modulus> std::fmt::Display for ModInt<M> {
        fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
            write!(f, "{}", self.0)
        }
    }
    impl<M: Modulus> PartialEq for ModInt<M> {
        fn eq(&self, rhs: &Self) -> bool {
            self.0 == rhs.0
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::Add<Rhs> for ModInt<M> {
        type Output = Self;
        fn add(self, rhs: Rhs) -> Self::Output {
            Self::from(self.0 + rhs.into().0)
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::AddAssign<Rhs> for ModInt<M> {
        fn add_assign(&mut self, rhs: Rhs) {
            *self = *self + rhs;
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::Sub<Rhs> for ModInt<M> {
        type Output = Self;
        fn sub(self, rhs: Rhs) -> Self::Output {
            Self::from(self.0 + M::VALUE - rhs.into().0)
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::SubAssign<Rhs> for ModInt<M> {
        fn sub_assign(&mut self, rhs: Rhs) {
            *self = *self - rhs;
        }
    }
    impl<M: Modulus> std::ops::Neg for ModInt<M> {
        type Output = Self;
        fn neg(self) -> Self::Output {
            Self::from(M::VALUE - self.0)
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::Mul<Rhs> for ModInt<M> {
        type Output = Self;
        fn mul(self, rhs: Rhs) -> Self::Output {
            Self::from(self.0 * rhs.into().0)
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::MulAssign<Rhs> for ModInt<M> {
        fn mul_assign(&mut self, rhs: Rhs) {
            *self = *self * rhs;
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::Div<Rhs> for ModInt<M> {
        type Output = Self;
        fn div(self, rhs: Rhs) -> Self::Output {
            self * num::pow(rhs.into(), M::VALUE as usize - 2)
        }
    }
    impl<M: Modulus, Rhs: Into<Self>> std::ops::DivAssign<Rhs> for ModInt<M> {
        fn div_assign(&mut self, rhs: Rhs) {
            *self = *self / rhs;
        }
    }
    impl<M: Modulus> num::traits::Zero for ModInt<M> {
        fn zero() -> Self {
            Self(0, PhantomData::<M>)
        }
        fn is_zero(&self) -> bool {
            self.0 == 0
        }
    }
    impl<M: Modulus> num::traits::One for ModInt<M> {
        fn one() -> Self {
            Self(1, PhantomData::<M>)
        }
    }
}