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 Go (1.14.1) (judge verdict: Accepted)

package main

import (
	"fmt"
)

const mod = 998244353

var ifac, fac []int

var pows map[int]map[int]int

func main() {
	fac = make([]int, 2e5+1)
	ifac = make([]int, 2e5+1)
	inv := make([]int, 2e5+1)
	fac[0], fac[1] = 1, 1
	ifac[0], ifac[1] = 1, 1
	inv[0], inv[1] = 1, 1
	for i := 2; i < 2e5+1; i++ {
		fac[i] = (fac[i-1] * i) % mod
		inv[i] = (mod - inv[mod%i]*(mod/i)) % mod
		if inv[i] < 0 {
			inv[i] += mod
		}
		ifac[i] = (ifac[i-1] * inv[i]) % mod
	}

	pows = make(map[int]map[int]int)
	var n, m, k int
	fmt.Scan(&n, &m, &k)
	var ans int
	for i := 0; i <= k; i++ {
		ans += (comb(n-1, i) * pow(m-1, n-1-i))
		ans %= mod
	}
	ans *= m
	ans %= mod
	fmt.Println(ans)
}

func comb(n, r int) int {
	if r > n {
		return 0
	} else if r == 0 || r == n {
		return 1
	}
	t := (ifac[n-r] * ifac[r]) % mod
	return (t * fac[n]) % mod
}

func pow(a, b int) int {
	if r, ok := pows[a][b]; ok {
		return r
	}
	if b == 0 {
		return 1
	}
	if b == 1 {
		return a
	}
	var r int
	if b%2 == 0 {
		r = pow(a, b/2) * pow(a, b/2)
		r %= mod
	} else {
		r = pow(a, b-1) * a
		r %= mod
	}
	if _, ok := pows[a]; !ok {
		pows[a] = make(map[int]int)
	}
	pows[a][b] = r
	return r
}

func min(a, b int) int {
	if a < b {
		return a
	}
	return b
}

func max(a, b int) int {
	if a > b {
		return a
	}
	return b
}

func abs(a int) int {
	if a < 0 {
		return -a
	}
	return a
}