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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

import Control.Monad
import Data.Maybe
import qualified Data.ByteString.Char8 as BS
import Data.List
import qualified Data.Vector as V

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine

main = do
    [n,m,k] <- getIntList
    print . sumMod $ map (\i -> mulMod (comb (n-1) i) (mulMod m  (powerMod (m-1) (n-1-i)))) [0..k]
    
moduloN :: Int
moduloN = 998244353

modN = flip mod moduloN

finv :: Int -> Int
finv k = finvs V.! k
    where finvs :: V.Vector Int
          finvs = V.generate (2 * 10 ^ 5 + 1) finv'
              where finv' 0 = 1
                    finv' n = modN $ finvs V.! (n - 1) * inv n

inv :: Int -> Int
inv k = invs V.! k
    where inv' 1 = 1
          inv' n = modN (-(invs V.! r) * q)
              where (q, r) = moduloN `divMod` n
          invs :: V.Vector Int
          invs = V.generate (2 * 10 ^ 5 + 1) inv'

fact :: Int -> Int
fact k = facts V.! k
    where fact' 0 = 1
          fact' n = modN $ n * (facts V.! (n - 1))
          facts :: V.Vector Int
          facts = V.generate (2 * 10 ^ 5 + 1) fact'

comb :: Int -> Int -> Int
comb n k | n < 0 || k < 0 || n < k = 0
         | otherwise = modN $ modN (fact n * finv k) * finv (n - k)

plusMod x y = modN $ x + y
sumMod xs = foldl' plusMod 0 xs

mulMod x y = modN $ x * y

powerMod x 0 = 1
powerMod x y
    | even y = mulMod t t
    | otherwise = mulMod x $ mulMod t t
    where
        t = powerMod x $ div y 2