Kth Number

AtCoder

IDabc295_e

Time4000ms

Memory256MB

Difficulty—

We have a sequence of length $N$ consisting of integers between $0$ and $M$, inclusive: $A=(A_1,A_2,\dots,A_N)$. Snuke will perform the following operations 1 and 2 in order. 1. For each $i$ such that $A_i=0$, independently choose a uniform random integer between $1$ and $M$, inclusive, and replace $A_i$ with that integer. 2. Sort $A$ in ascending order. Print the expected value of $A_K$ after the two operations, modulo $998244353$. How to print a number modulo $998244353$? It can be proved that the sought expected value is always rational. Additionally, under the Constraints of this problem, when that value is represented as $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, it can be proved that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Print this $R$. ## Constraints * $1\leq K \leq N \leq 2000$ * $1\leq M \leq 2000$ * $0\leq A_i \leq M$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $K$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

3 5 2
2 0 4

Output #1

3

In the operation 1, Snuke will replace $A_2$ with an integer between $1$ and $5$. Let $x$ be this integer.

*   If $x=1$ or $2$, we will have $A_2=2$ after the two operations.
*   If $x=3$, we will have $A_2=3$ after the two operations.
*   If $x=4$ or $5$, we will have $A_2=4$ after the two operations.

Thus, the expected value of $A_2$ is $\frac{2+2+3+4+4}{5}=3$.

Input #2

2 3 1
0 0

Output #2

221832080

The expected value is $\frac{14}{9}$.

Input #3

10 20 7
6 5 0 2 0 0 0 15 0 0

Output #3

617586310

API Response (JSON)

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  "problem": {
    "name": "Kth Number",
    "description": {
      "content": "We have a sequence of length $N$ consisting of integers between $0$ and $M$, inclusive: $A=(A_1,A_2,\\dots,A_N)$. Snuke will perform the following operations 1 and 2 in order. 1.  For each $i$ such th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc295_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a sequence of length $N$ consisting of integers between $0$ and $M$, inclusive: $A=(A_1,A_2,\\dots,A_N)$.\nSnuke will perform the following operations 1 and 2 in order.\n\n1.  For each $i$ such th...",
      "is_translate": false,
      "language": "English"
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}

Full JSON Raw Segments