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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
using static System.Math;

namespace CSharpSpiral {
	class Program {
		static void Main(string[] args) {
			var sw = new StreamWriter(Console.OpenStandardOutput()) {
				AutoFlush=
#if DEBUG
			true
#else
			false
#endif
			};
			Console.SetOut(sw);
			new Program().Solve(new ConsoleInput(
#if DEBUG
			new StreamReader("../../../inputData.txt")
#else
			Console.In
#endif
			, ' '));
			Console.Out.Flush();
		}
		public void Solve(ConsoleInput cin) {
			var F = new ResidueField(998244353);
			var N = cin.ReadInt;
			var M = cin.ReadInt;
			var K = cin.ReadInt;
			var Sum = K*2<N-1
			? Enumerable.Range(0, K+1).Sum(j => F.Calculate(M, N-1, j, 'C', M-1, N-1-j, '^', '*', '*'))
			: F.Pow(M, N)
				-Enumerable.Range(0, N-K-1).Sum(jj => F.Calculate(M, N-1, jj, 'C', M-1, jj, '^', '*', '*'));
			var Res = F.mod(Sum);
			Console.WriteLine(Res);
		}
	}
	static class ExtendClass {
		public static string Join<T>(this IEnumerable<T> X, string separator = " ") => string.Join(separator, X);
	}
	class ConsoleInput {
		private readonly TextReader _stream;
		private readonly char _separator = ' ';
		private readonly Queue<string> inputStream;
		public ConsoleInput(TextReader stream, char separator = ' ') {
			this._separator=separator;
			this._stream=stream;
			inputStream=new Queue<string>();
		}
		public string Read {
			get {
				if(inputStream.Count!=0) return inputStream.Dequeue();
				string[] tmp = _stream.ReadLine().Split(_separator);
				for(int i = 0; i<tmp.Length; ++i)
					inputStream.Enqueue(tmp[i]);
				return inputStream.Dequeue();
			}
		}
		public string ReadLine { get { return _stream.ReadLine(); } }
		public int ReadInt { get { return int.Parse(Read); } }
		public long ReadLong { get { return long.Parse(Read); } }
		public double ReadDouble { get { return double.Parse(Read); } }
		public string[] ReadStrArray(long N) {
			var ret = new string[N];
			for(long i = 0; i<N; ++i)
				ret[i]=Read;
			return ret;
		}
		public int[] ReadIntArray(long N) { var ret = new int[N]; for(long i = 0; i<N; ++i) ret[i]=ReadInt; return ret; }
		public long[] ReadLongArray(long N) { var ret = new long[N]; for(long i = 0; i<N; ++i) ret[i]=ReadLong; return ret; }
		public double[] ReadDoubleArray(long N) { var ret = new double[N]; for(long i = 0; i<N; ++i) ret[i]=ReadDouble; return ret; }
		public bool isEOF {
			get { return _stream.Peek()==-1; }
		}
	}
	class ResidueField: QuotientRing {
		public ResidueField(int _modulo = 1000*1000*1000+7) : base(_modulo) { }
		private static long extgcd(long a, long b, ref long x, ref long y) {
			if(b==0) {
				x=1; y=0; return a;
			}
			var q = a/b;
			var g = extgcd(b, a-q*b, ref x, ref y);
			var z = x-q*y;
			x=y; y=z;
			return g;
		}
		public long Inv(long a) {
			if(a==0) throw new DivideByZeroException();
			long x = 0, y = 0;
			extgcd(a, modulo, ref x, ref y);
			return mod(x);
		}
		public long Div(long a, long b) => Mul(a, Inv(b));
		public override long Permutation(int n, int k) {
			if(n<=factorialTableLastIndex) return Div(factorialTable[n], factorialTable[n-k]);
			else if(n<=500000) return Div(Factorial(n), Factorial(n-k));
			else return base.Permutation(n, k);
		}
		public long Combination(int n, int k) {
			if(k<0||k>n) return 0;
			if(n-k<k) k=n-k;
			return Div(Permutation(n, k), Factorial(k));
		}
		public override long Calculate(params object[] RPN) {
			var S = new Stack<long>();
			foreach(var token in RPN) {
				var type = token.GetType();
				if(type==typeof(int)) {
					S.Push((int)token);
				} else if(type==typeof(long)) {
					S.Push((long)token);
				} else if(type==typeof(char)) {
					long v1 = 0L, v2 = 0L;
					v2=S.Pop();
					HashSet<char> unaries = new HashSet<char>() { '!' };
					if(!unaries.Contains((char)token))
						v1=S.Pop();
					long res;
					switch((char)token) {
						case '+': res=Add(v1, v2); break;
						case '-': res=Sub(v1, v2); break;
						case '*': res=Mul(v1, v2); break;
						case '/': res=Div(v1, v2); break;
						case '^': res=Pow(v1, v2); break;
						case '!': res=Factorial((int)v2); break;
						case 'C': res=Combination((int)v1, (int)v2); break;
						case 'P': res=Permutation((int)v1, (int)v2); break;
						default: throw new InvalidOperationException();
					}
					S.Push(res);
				} else throw new InvalidOperationException();
			}
			return S.Peek();
		}
	}
	class QuotientRing {
		public readonly long modulo;
		public QuotientRing(int mModulo = 1000*1000*1000+7) {
			modulo=mModulo;
		}
		protected long[] factorialTable = { 1 };
		protected int factorialTableLastIndex { get { return factorialTable.Length-1; } }
		public long mod(long x) => x<0 ? x%modulo+modulo : x%modulo;
		public long Add(long a, long b) => mod(a+b);
		public long Sub(long a, long b) => mod(a-b);
		public long Mul(long a, long b) => mod(a*b);
		public long Pow(long a, long n) {
			if(n==0) return 1;
			var R = 1L;
			n=n<0 ? n%(modulo-1)+(modulo-1) : n%(modulo-1);
			for(; n>0; a=Mul(a, a), n>>=1) {
				if((n&1)!=0) R=Mul(R, a);
			}
			return R;
		}
		private void ExpandFactorialTable(int N) {
			var k = factorialTableLastIndex;
			Array.Resize(ref factorialTable, N+1);
			for(int i = k+1; i<=N; i++) {
				if(i==0) { factorialTable[i]=1; continue; }
				factorialTable[i]=Mul(factorialTable[i-1], i);
			}
		}
		public long Factorial(int n) {
			if(n>factorialTableLastIndex)
				ExpandFactorialTable(n);
			return factorialTable[n];
		}
		public virtual long Permutation(int n, int k) {
			if(k<0||k>n) return 0;
			var Res = 1L;
			for(int i = 0; i<k; i++)
				Res=Mul(Res, n-i);
			return Res;
		}
		public virtual long Calculate(params object[] RPN) {
			var S = new Stack<long>();
			foreach(var token in RPN) {
				var type = token.GetType();
				if(type==typeof(int)||type==typeof(long)) {
					S.Push((long)token);
				} else if(type==typeof(char)) {
					long v1 = 0L, v2 = 0L;
					v2=S.Pop();
					HashSet<char> unaries = new HashSet<char>() { '!' };
					if(!unaries.Contains((char)token))
						v1=S.Pop();
					long res;
					switch((char)token) {
						case '+': res=Add(v1, v2); break;
						case '-': res=Sub(v1, v2); break;
						case '*': res=Mul(v1, v2); break;
						case '^': res=Pow(v1, v2); break;
						case '!': res=Factorial((int)v2); break;
						default: throw new InvalidOperationException();
					}
					S.Push(res);
				} else throw new InvalidOperationException();
			}
			return S.Peek();
		}
	}
}