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 C++ (GCC 9.2.1) (judge verdict: Time Limit Exceeded)

#include<bits/stdc++.h>
using namespace std;

typedef long long ll;

const ll M = 998244353;

vector<ll> fac(300001); 
vector<ll> ifac(300001); 

ll mpow(ll x, ll n){ 
    ll ans = 1;
    while(n != 0){
        if(n&1) ans = ans*x % M;
        x = x*x % M;
        n = n >> 1;
    }
    return ans;
}

ll comb(ll a, ll b){ //aCbをmod計算
    if(a == 0 && b == 0)return 1;
    if(a < b || a < 0)return 0;
    ll tmp = ifac[a-b]* ifac[b] % M;
    return tmp * fac[a] % M;
}

int main(){
    ll N,X,K;
    cin>>N>>X>>K;

    fac[0] = 1;
    ifac[0] = 1;
    for(ll i = 0; i<300000; i++){
        fac[i+1] = fac[i]*(i+1) % M; // n!(mod M)
        ifac[i+1] = ifac[i]*mpow(i+1, M-2) % M;
    }


    ll ans=0;
    for(long i=0;i<=K;i++){
      long Z=X;
      Z*=comb(N-1,i);
      Z%=M;
      for(long j=0;j<N-1-i;j++){
        Z*=X-1;
        Z%=M;
      }
      ans+=Z;
      ans%=M;
    }
    cout << ans << endl;
    return 0;
}