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G - Gardens

Score : $600$ points

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.

2 2 1 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.)

2 0 0 0

0

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

2 0 2 2

9

100 12 34 56

769445780

5000000 2521993 2967363 3802088

264705492