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)

N, M, K = gets.split.map(&:to_i)
MOD = 998244353

def mpow(x, n)
  ret = 1
  while n > 0
    if n.even?
      x = x * x % MOD
      n >>= 1
    else
      ret = ret * x % MOD
      n -= 1
    end
  end
  ret
end

@fac, @inv, @fiv = [1, 1], [1, 1], [1, 1]
(2 ... N).each do |i|
  @fac[i] = @fac[i - 1] * i % MOD
  @inv[i] = MOD - @inv[MOD % i] * (MOD / i) % MOD
  @fiv[i] = @fiv[i - 1] * @inv[i] % MOD
end
c = ->(n, m){@fac[n] * @fiv[n - m] * @fiv[m] % MOD}
f = ->(n, m, i){m * c[n - 1, i] * mpow(M - 1, N - 1 - i) % MOD}
puts (0 .. K).inject(0){|s, i| (s + f[N, M, i]) % MOD}