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 Python (3.8.2) (judge verdict: Accepted)


def pow_mod(a, b, m):
    if b == 0:
        return 1
    elif b % 2 == 0:
        d = pow_mod(a, b//2, m)
        return (d * d) % m
    else:
        return (a * pow_mod(a,b-1,m))%m

def comb(n, c, m):
    upe = 1
    dow = 1
    for i in range(1,c+1):
        upe = upe * n % m
        dow = dow * i % m
        n -= 1
    return upe * pow_mod(dow, m-2,m)%m

def modinv(a, m):
    b = m
    u = 1
    v = 0
    while b:
        t = a // b
        a -= t * b
        a, b = b, a
        u -= t * v
        u, v = v, u
    u %= m
    if u < 0:
        u += m
    return u

MOD = 998244353
def solve(N, M, K):
    MOD = 998244353
    U = N-1-K
    color = M * pow_mod(M-1, U, MOD) % MOD
    c = comb(N-1, U, MOD) % MOD
    #print("c0=%d"%c)
    answer = (c * color) % MOD
    #print("a0", answer)
    #print("U=%d"%U)
    for j in range(N-K, N):
        c = ((c * (N-j)%MOD) * modinv(j, MOD)) % MOD
        #assert c == comb(N-1, j, MOD)
        color = (color * (M-1)) % MOD
        a = (c * color) % MOD
        answer += a
    return answer % MOD
N, M, K = map(int, input().split())
print(solve(N, M, K))