Greedy Division

AtCoder

IDagc054_b

Time2000ms

Memory256MB

Difficulty—

We have $N$ oranges, numbered $1$ through $N$. The weight of Orange $i$ is $W_i$. Takahashi and Aoki will share these oranges as follows. * Choose a permutation $(p_1, p_2, \cdots, p_N)$ of $(1,2,\cdots,N)$. * For each $i = 1, 2, \cdots, N$ in this order, do the following. * If the total weight of the oranges Takahashi has taken is not greater than that of the oranges Aoki has taken, Takahashi takes Orange $p_i$. Otherwise, Aoki takes Orange $p_i$. Find the number of permutations $p$ modulo $998244353$ such that the total weight of the oranges Takahashi will take is equal to that of the oranges Aoki will take. ## Constraints * $2 \leq N \leq 100$ * $1 \leq W_i \leq 100$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $W_1$ $W_2$ $\cdots$ $W_N$ [samples]

Samples

Input #1

3
1 1 2

Output #1

4

There are four permutations $p$ satisfying the condition: $(1,3,2),(2,3,1),(3,1,2),(3,2,1)$. If $p=(3,2,1)$, for example, the following will happen.

*   $i=1$: the total weights of the oranges Takahashi and Aoki have taken are $0$ and $0$, respectively. Takahashi takes Orange $p_i=3$.
*   $i=2$: the total weights of the oranges Takahashi and Aoki have taken are $2$ and $0$, respectively. Aoki takes Orange $p_i=2$.
*   $i=3$: the total weights of the oranges Takahashi and Aoki have taken are $2$ and $1$, respectively. Aoki takes Orange $p_i=1$.

Thus, the permutation $p=(3,2,1)$ counts.

Input #2

4
1 2 3 8

Output #2

Input #3

20
2 8 4 7 5 3 1 2 4 1 2 5 4 3 3 8 1 7 8 2

Output #3

373835282

API Response (JSON)

{
  "problem": {
    "name": "Greedy Division",
    "description": {
      "content": "We have $N$ oranges, numbered $1$ through $N$. The weight of Orange $i$ is $W_i$. Takahashi and Aoki will share these oranges as follows. *   Choose a permutation $(p_1, p_2, \\cdots, p_N)$ of $(1,2,\\",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc054_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have $N$ oranges, numbered $1$ through $N$. The weight of Orange $i$ is $W_i$. Takahashi and Aoki will share these oranges as follows.\n\n*   Choose a permutation $(p_1, p_2, \\cdots, p_N)$ of $(1,2,\\...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments