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 Fortran (GNU Fortran 9.2.1) (judge verdict: Accepted)

program main
    use,intrinsic :: iso_fortran_env
    implicit none
    integer(int64):: n,m,k,ans,i,c,f
    integer(int64), allocatable:: frac(:), ifrac(:)
    integer(int64),parameter:: md = 998244353

    read*, n,m,k

    allocate(frac(0:n), ifrac(0:n))
    frac(0) = 1
    ifrac(0) = 1
    do i=1,n
        frac(i) = mod(frac(i-1)*i,md)
        ifrac(i) = inv(frac(i),md)
    end do

    do i=0,k
        f = mod(m*factorial(m-1,n-i-1,md),md)
        c = comb(n-1,i,md)
        ans = mod(ans + mod(f*c,md),md)
    end do

    print'(i0)', ans
contains
    function comb(a,b,md) result(ret)
        integer(int64):: a,b,md
        integer(int64):: ret
        ret = mod(mod(frac(a)*ifrac(b),md)*ifrac(a-b),md)
    end function

    function inv(x,md) result(ret)
        integer(int64),intent(in):: x,md
        integer(int64):: ret
        ret = factorial(x,md-2,md)
    end function

    function factorial(a,b,md) result(ret)
        integer(int64):: a,b,md
        integer(int64):: aa,bb,ret
        aa = a
        bb = b
        ret = 1
        do while(bb > 0)
            if (mod(bb,2_8)/=0) ret=mod(ret*aa,md)
            aa=mod(aa*aa,md)
            bb=bb/2
        end do
    end function
end program main