Sugoroku 4

AtCoder

IDabc275_e

Time2000ms

Memory256MB

Difficulty—

Takahashi is playing sugoroku, a board game. The board has $N+1$ squares, numbered $0$ to $N$. Takahashi starts at square $0$ and goes for square $N$. The game uses a roulette wheel with $M$ numbers from $1$ to $M$ that appear with equal probability. Takahashi spins the wheel and moves by the number of squares indicated by the wheel. If this would send him beyond square $N$, he turns around at square $N$ and goes back by the excessive number of squares. For instance, assume that $N=4$ and Takahashi is at square $3$. If the wheel shows $4$, the excessive number of squares beyond square $4$ is $3+4-4=3$. Thus, he goes back by three squares from square $4$ and arrives at square $1$. When Takahashi arrives at square $N$, he wins and the game ends. Find the probability, modulo $998244353$, that Takahashi wins when he may spin the wheel at most $K$ times. How to print a probability modulo $998244353$It can be proved that the sought probability is always a rational number. Additionally, under the Constraints of this problem, when the sought probability is represented as an irreducible fraction $\frac{y}{x}$, it is guaranteed that $x$ is not divisible by $998244353$. Here, there is a unique integer $z$ between $0$ and $998244352$ such that $xz \equiv y \pmod{998244353}$. Print this $z$. ## Constraints * $M \leq N \leq 1000$ * $1 \leq M \leq 10$ * $1 \leq K \leq 1000$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $K$ [samples]

Samples

Input #1

2 2 1

Output #1

499122177

Takahashi wins in one spin if the wheel shows $2$. Therefore, the probability of winning is $\frac{1}{2}$.
We have $2\times 499122177 \equiv 1 \pmod{998244353}$, so the answer to be printed is $499122177$.

Input #2

10 5 6

Output #2

184124175

Input #3

100 1 99

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Sugoroku 4",
    "description": {
      "content": "Takahashi is playing sugoroku, a board game. The board has $N+1$ squares, numbered $0$ to $N$. Takahashi starts at square $0$ and goes for square $N$. The game uses a roulette wheel with $M$ numbers f",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc275_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi is playing sugoroku, a board game.\nThe board has $N+1$ squares, numbered $0$ to $N$. Takahashi starts at square $0$ and goes for square $N$.\nThe game uses a roulette wheel with $M$ numbers f...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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