Collect Them All

AtCoder

IDabc331_g

Time2000ms

Memory256MB

Difficulty—

There are $N$ cards in a box. Each card has an integer written on it, which is between $1$ and $M$, inclusive. For each $i=1,\ldots,M$, there are $C_i$ cards with the number $i$ written on them. Starting with an empty notebook, you repeat the following operation: * Randomly draw one card from the box. Write the integer on the card in the notebook, then return the card to the box. Find the expected value, modulo $998244353$, of the number of times the operation is performed until the notebook has all integers from $1$ to $M$ written at least once. How to find an expected value modulo $998244353$ The expected value sought in this problem can be proven to always be a rational number. Furthermore, the constraints of this problem guarantee that when the expected value is expressed as an irreducible fraction $\frac yx$, the denominator $x$ is not divisible by $998244353$. Then, there is a unique $0\leq z\lt998244353$ such that $y\equiv xz\pmod{998244353}$. Print this $z$. ## Constraints * $1 \leq M \leq N \leq 2\times 10^5$ * $1 \leq C_i$ * $\sum_{i=1}^{M}C_i=N$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $C_1$ $\ldots$ $C_M$ [samples]

Samples

Input #1

2 2
1 1

Output #1

3

The series of operations may proceed as follows:

*   You draw a card with $1$ written on it. The notebook now has one $1$ written in it.
*   You again draw a card with $1$ written on it. The notebook now has two $1$s written in it.
*   You draw a card with $2$ written on it. The notebook now has two $1$s and one $2$ written in it.

The sought expected value is $2\times\frac{1}{2}+3\times\frac{1}{4}+4\times\frac{1}{8}+\ldots=3$.

Input #2

5 2
4 1

Output #2

748683270

The expected value is $\frac{21}{4}$, whose representation modulo $998244353$ is $748683270$.

Input #3

50 50
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Output #3

244742906

The expected value is $\frac{13943237577224054960759}{61980890084919934128}$.

Input #4

74070 15
1 2 3 11 22 33 111 222 333 1111 2222 3333 11111 22222 33333

Output #4

918012973

API Response (JSON)

{
  "problem": {
    "name": "Collect Them All",
    "description": {
      "content": "There are $N$ cards in a box. Each card has an integer written on it, which is between $1$ and $M$, inclusive. For each $i=1,\\ldots,M$, there are $C_i$ cards with the number $i$ written on them. Start",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc331_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ cards in a box. Each card has an integer written on it, which is between $1$ and $M$, inclusive. For each $i=1,\\ldots,M$, there are $C_i$ cards with the number $i$ written on them.\nStart...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments