Permutation

AtCoder

IDabc199_e

Time2000ms

Memory256MB

Difficulty—

Print the number of sequences $a$ that are permutations of $(1, 2, 3, \dots, N)$ and satisfy the following condition: * for every integer $i$ such that $1 \le i \le M$, at most $Z_i$ numbers among $a_1, a_2, a_3, \dots, a_{X_i}$ are less than or equal to $Y_i$ . ## Constraints * $2 \le N \le 18$ * $0 \le M \le 100$ * $1 \le X_i \lt N$ * $1 \le Y_i \lt N$ * $0 \le Z_i \lt N$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $X_1$ $Y_1$ $Z_1$ $X_2$ $Y_2$ $Z_2$ $X_3$ $Y_3$ $Z_3$ $\hspace{23pt} \vdots$ $X_M$ $Y_M$ $Z_M$ [samples]

Samples

Input #1

3 1
2 2 1

Output #1

4

The four sequences $a$ satisfying the condition are:

*   $(1, 3, 2)$
*   $(2, 3, 1)$
*   $(3, 1, 2)$
*   $(3, 2, 1)$

$(1, 2, 3)$ and $(2, 1, 3)$ violate the condition, since each of them has two numbers less than or equal to $2$ among $a_1, a_2$.

Input #2

5 2
3 3 2
4 4 3

Output #2

Input #3

18 0

Output #3

6402373705728000

API Response (JSON)

{
  "problem": {
    "name": "Permutation",
    "description": {
      "content": "Print the number of sequences $a$ that are permutations of $(1, 2, 3, \\dots, N)$ and satisfy the following condition: *   for every integer $i$ such that $1 \\le i \\le M$, at most $Z_i$ numbers among ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc199_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Print the number of sequences $a$ that are permutations of $(1, 2, 3, \\dots, N)$ and satisfy the following condition:\n\n*   for every integer $i$ such that $1 \\le i \\le M$, at most $Z_i$ numbers among ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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