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 Ruby (2.7.1) (judge verdict: Accepted)

require 'set'

class Hash
  def push(key, value)
    self[key] = [] if self[key] == nil
    self[key].push(value)
  end
end

class Array
  def lower_bound(value)
    left = -1;
    right = self.length;
    while left + 1 < right
      mid = left + (right - left) / 2;
      if self[mid] >= value
        right = mid
      else
        left = mid
      end
    end
    right
  end

  def unique
    res = [self.first]
    each_cons(2) do |a, b|
      if a != b
        res.push(b)
      end
    end
    res
  end
end

def get_ints
  gets.split.map(&:to_i)
end

def get_ints_minus_one
  get_ints.map { |x| x - 1 }
end

def get_int
  gets.chomp.to_i
end

class Combination
  def initialize(n, mod)
    @mod = mod
    @fac, @finv, @inv = Array.new(3) { [] }
    @fac[0] = @fac[1] = 1
    @finv[0] = @finv[1] = 1
    @inv[1] = 1
    i = 2
    while i < n
      @fac[i] = @fac[i - 1] * i % @mod
      @inv[i] = @mod - @inv[@mod % i] * (@mod / i) % @mod
      @finv[i] = @finv[i - 1] * @inv[i] % @mod
      i += 1
    end
  end

  def nCk(n, k)
    return 0 if (n < k) || (n < 0 || k < 0)

    @fac[n] * (@finv[k] * @finv[n - k] % @mod) % @mod
  end
end

N, M, K = get_ints
MOD = 998244353

ans = 0

m_stock = [1]

test = 1
1.step(N) do |i|
  test = (test * (M-1)) % MOD
  m_stock[i] = test
end

combination = Combination.new(N, MOD)

0.step(K) do |k|
  stock = M * m_stock[N-k-1] % MOD
  stock *= combination.nCk(N-1, k)
  stock %= MOD
  ans += stock
  ans %= MOD
end

puts ans