Description

Now three people are playing Dou Dizhu. If the Landlord achieves Spring, then the Landlord is considered the winner; otherwise, even if the Landlord plays all their cards first, it is still considered a win for the Farmers. Assume all three players act with optimal decisions.

You are given $n(0 \leq n\leq 20)$ cards. Ask how many possible initial hands of the Landlord contain these $n$ cards, and no matter how the Farmers’ cards are distributed, the Landlord is guaranteed to achieve Spring.

Input Format

The first line contains an integer $t$ indicating the number of test cases.

For each test case, one line is given. The first integer is $n$, the number of fixed cards, followed by $n$ space-separated integers describing each fixed card.

In particular, we use $1$ to represent rank A, $11$ for rank J, $12$ for rank Q, $13$ for rank K, $14$ for the small Joker, and $15$ for the big Joker. The input is guaranteed to be valid, meaning that the count of each rank will not exceed its count in a full deck.

Output Format

For each test case, output one integer, the number of Landlord hands that satisfy the condition. The answer may be very large; output it modulo $998244353$.

Note: In this problem we ignore suits. If two hands have exactly the same multiset of ranks but different suits, they are considered the same.

6
20 1 2 2 3 4 5 6 7 8 8 9 10 11 11 12 13 13 13 13 15
20 1 1 2 2 3 4 5 6 7 8 9 10 11 12 12 12 12 13 13 14
20 1 2 2 3 3 4 5 6 7 7 7 7 8 9 10 10 11 12 13 15
20 1 2 3 4 4 5 6 7 8 9 10 11 11 12 13 13 13 13 14 15
3 3 3 3
4 3 3 3 3

1
0
1
1
4790
1670

Hint

Sample Explanation

For the first sample, we can see that the Farmers cannot have a Bomb or Rocket, so the Landlord can first play $[3,4,5,6,7,8,9,10,J,Q]$ (clearly neither Farmer can beat it), then play $[2,2]$, then play the big Joker, then play $[K,K,K,K,A,J]$, and finally play $[8]$.

Translated by ChatGPT 5