Score : $625$ points

问题描述

在一个盒子里有 $N$ 张卡片。每张卡片上都写有一个整数，这个整数介于 $1$ 和 $M$（包含）之间。对于每个 $i=1,\ldots,M$，有 $C_i$ 张卡片上写着数字 $i$。

从一本空白笔记本开始，你重复以下操作：

• 随机从盒子里抽出一张卡片。将卡片上的整数记在笔记本上，然后将卡片放回盒子。

求出执行该操作直至笔记本中至少记录过一次从 $1$ 到 $M$ 的所有整数的期望次数，结果对 $998244353$ 取模。

如何求解模 $998244353$ 下的期望值 本问题所求的期望值可以被证明始终是一个有理数。此外，本题的约束条件保证了当期望值表示为不可约分数 $\frac yx$ 时，分母 $x$ 不会是 $998244353$ 的倍数。因此，存在一个唯一的 $0\leq z\lt998244353$，使得 $y\equiv xz\pmod{998244353}$。请输出这个 $z$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$N$ $M$

$C_1$ $\ldots$ $C_M$

Output

Print the answer.

Sample Input 1

2 2
1 1

Sample 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 $2\times\frac{1}{2}+3\times\frac{1}{4}+4\times\frac{1}{8}+\ldots=3$.

Sample Input 2

5 2
4 1

Sample Output 2

748683270

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

Sample 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

Sample Output 3

244742906

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

Sample Input 4

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

Sample Output 4

918012973

update @ 2024/3/10 01:16:32