AtCoder Language

AtCoder

IDarc172_b

Time2000ms

Memory256MB

Difficulty—

The AtCoder language has $L$ different characters. How many $N$\-character strings $s$ consisting of characters in the AtCoder language satisfy the following condition? Find the count modulo $998244353$. * All $K$\-character subsequences of the string $s$ are different. More precisely, there are $_N\mathrm{C}_K$ ways to obtain a $K$\-character string by extracting $K$ characters from the string $s$ and concatenating them without changing the order, and all of these ways must generate different strings. What is $_N\mathrm{C}_K$?$_N\mathrm{C}_K$ refers to the total number of ways to choose $K$ from $N$ items. More precisely, $_N\mathrm{C}_K$ is the value of $N!$ divided by $K! \times (N-K)!$. ## Constraints * $1 \leq K < N \leq 500000$ * $1 \leq L \leq 10^9$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ $L$ [samples]

Samples

Input #1

4 3 2

Output #1

2

If `a` and `b` represent the first and second characters in the language, the condition is satisfied by two strings: `abab` and `baba`.

Input #2

100 80 26

Output #2

496798269

Approximately $10^{86}$ strings satisfy the condition, but here we print the count modulo $998244353$, which is $496798269$.

Input #3

100 1 26

Output #3

Input #4

500000 172172 503746693

Output #4

API Response (JSON)

{
  "problem": {
    "name": "AtCoder Language",
    "description": {
      "content": "The AtCoder language has $L$ different characters. How many $N$\\-character strings $s$ consisting of characters in the AtCoder language satisfy the following condition? Find the count modulo $99824435",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc172_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "The AtCoder language has $L$ different characters. How many $N$\\-character strings $s$ consisting of characters in the AtCoder language satisfy the following condition? Find the count modulo $99824435...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments