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"

type Combination struct {
	n int
	f *Factorial
}

func NewCombination(n int, m int64) *Combination {
	f := NewFactorial(n, m)
	return &Combination{n, f}
}

func (c *Combination) Combination(n, k int) int64 {
	return c.f.F[n] * c.f.Inv[k] % c.f.Mod * c.f.Inv[n-k] % c.f.Mod
}

func (c *Combination) Permutation(n, k int) int64 {
	return c.Combination(n+k-1, n-1)
}

func (c *Combination) Factorial(n int) int64 {
	return c.f.F[n]
}

type Factorial struct {
	n      int
	Mod    int64
	F, Inv []int64
}

func NewFactorial(n int, m int64) *Factorial {
	f := Factorial{n, m, nil, nil}
	f.F = make([]int64, n+1)
	f.Inv = make([]int64, n+1)
	f.F[0] = 1
	for i := 1; i <= n; i++ {
		f.F[i] = f.F[i-1] * int64(i) % f.Mod
	}
	f.Inv[n] = pow(f.F[n], f.Mod-2, f.Mod)
	for i := n; i > 0; i-- {
		f.Inv[i-1] = f.Inv[i] * int64(i) % f.Mod
	}
	return &f
}

func pow(x, n, m int64) int64 {
	if n == 0 {
		return 1
	}
	if n&1 == 0 {
		r := pow(x, n/2, m)
		return r * r % m
	}
	return x * pow(x, n-1, m) % m
}

func div(x, n, m int64) int64 {
	return x * pow(n, m-2, m) % m
}

func main() {
	var n, m, k int
	const M int64 = 998244353
	fmt.Scan(&n, &m, &k)
	c := NewCombination(n, M)
	var ans int64
	for i := 0; i <= k; i++ {
		m64, p := int64(m), int64(n-i-1)
		x := c.Combination(n-1, i)
		x *= m64 * pow(m64-1, p, M) % M
		x %= M
		ans += x
		ans %= M
	}
	fmt.Println(ans)
}