Score : $600$ points

问题陈述

Takahashi 有 $A$ 棵苹果树苗，$B$ 棵香蕉树苗和 $C$ 棵樱桃树苗。同种类的树苗之间不可区分。 他将在他的 $N$ 个花园中种植这些树苗，以满足所有以下条件。

• 每个花园中至少要种植一棵树苗。
• 同一个花园中不允许种植两棵或更多棵同种类的树苗。
• 不需要种植他所拥有的所有树苗。

有多少种方式可以种植树苗以满足这些条件？请在 $998244353$ 取模后找到方案数。 两种方式被认为是不同的，当存在这样一个花园，在这两种方式中有不同组合的树苗被种植。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Takahashi has $A$ apple seedlings, $B$ banana seedlings, and $C$ cherry seedlings. Seedlings of the same kind cannot be distinguished.
He will plant these seedlings in his $N$ gardens so that all of the following conditions are satisfied.

• At least one seedling must be planted in every garden.
• It is not allowed to plant two or more seedlings of the same kind in the same garden.
• It is not necessary to plant all seedlings he has.

How many ways are there to plant seedlings to satisfy the conditions? Find the count modulo $998244353$.
Two ways are distinguished when there is a garden with different sets of seedlings planted in these two ways.

Constraints

• $1 \leq N \leq 5 \times 10^6$
• $0 \leq A \leq N$
• $0 \leq B \leq N$
• $0 \leq C \leq N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $A$ $B$ $C$

Output

Print the answer.

Sample Input 1

2 2 1 1

Sample Output 1

21

As illustrated below, there are $21$ ways to plant seedlings to satisfy the conditions.
(The two frames arranged vertically are the gardens. $A, B, C$ stand for apple, banana, cherry, respectively.)

Sample Input 2

2 0 0 0

Sample Output 2

0

There may be no way to plant seedlings to satisfy the conditions.

Sample Input 3

2 0 2 2

Sample Output 3

9

Sample Input 4

100 12 34 56

Sample Output 4

769445780

Sample Input 5

5000000 2521993 2967363 3802088

Sample Output 5

264705492

update @ 2024/3/10 10:12:45