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.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest.ABC_167
{
    class E
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            long[] nmk = ReadLongs();
            long n = nmk[0];
            long m = nmk[1];
            long k = nmk[2];
            if (n == 1)
            {
                WriteLine(m);
                return;
            }

            long mask = 998244353;
            CaseCalculator calculator = new CaseCalculator(mask, n);

            long res = 0;
            long[] pows = new long[n];
            pows[0] = 1;
            for(int i = 1; i < n; i++)
            {
                pows[i] = pows[i - 1] * (m - 1);
                pows[i] %= mask;
            }
            for (long i = n - 1; n - 1 - i <= k; i--)
            {
                long tmp = m * pows[i];
                tmp %= mask;
                tmp *= calculator.Combination(n - 1, i);
                tmp %= mask;
                res += tmp;
                res %= mask;
            }
            WriteLine(res);
        }

        class CaseCalculator
        {
            long mask;
            long[] allPermutations;
            long[] allInverses;

            public CaseCalculator(long mask, long permutationCnt)
            {
                this.mask = mask;
                allPermutations = AllPermutations(permutationCnt);
                allInverses = AllInverses(permutationCnt);
            }

            public long Combination(long n, long m)
            {
                if (n < m) return 0;

                if (n - m < m) m = n - m;

                return Multi(allPermutations[n], allInverses[n - m], allInverses[m]);
            }

            public long Permutation(long n, long m)
            {
                if (n < m) return 0;

                return Multi(allPermutations[n], allInverses[n - m]);
            }

            long[] AllPermutations(long n)
            {
                var perms = new long[n + 1];
                perms[0] = 1;
                for (int i = 1; i <= n; i++)
                {
                    perms[i] = Multi(perms[i - 1], i);
                }
                return perms;
            }

            long[] AllInverses(long n)
            {
                var inverses = new long[n + 1];
                inverses[1] = 1;
                var permInverses = new long[n + 1];
                permInverses[0] = 1;
                permInverses[1] = 1;
                for (int i = 2; i <= n; i++)
                {
                    inverses[i] = mask - inverses[mask % i] * (mask / i) % mask;
                    permInverses[i] = Multi(permInverses[i - 1], inverses[i]);
                }
                return permInverses;
            }

            public long Multi(params long[] vals)
            {
                if (vals.Length == 0)
                {
                    return 0;
                }

                long val = vals[0] % mask;
                for (int i = 1; i < vals.Length; i++)
                {
                    val *= (vals[i] % mask);
                    val %= mask;
                }
                return val;
            }

            public long Inverse(long val)
            {
                long a = mask;
                long b = val % mask;
                long x = 1;
                long x1 = 0;
                long y = 0;
                long y1 = 1;
                while (b > 0)
                {
                    long q = a / b;
                    long r = a % b;
                    long x2 = (mask + x - q * x1) % mask;
                    long y2 = (mask + y - q * y1) % mask;
                    a = b;
                    b = r;
                    x = x1;
                    x1 = x2;
                    y = y1;
                    y1 = y2;
                }
                return y;
            }

            long DeprecatedInverse(long val)
            {
                return Pow(val, mask - 2);
            }

            long Pow(long a, long exp)
            {
                if (exp == 0) return 1;
                else if (exp == 1) return a % mask;
                else
                {
                    long halfMod = Pow(a, exp / 2);
                    long nextMod = Multi(halfMod, halfMod);
                    if (exp % 2 == 0)
                    {
                        return nextMod;
                    }
                    else
                    {
                        return Multi(nextMod, a);
                    }
                }
            }
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}