Bessie and Cards
Description
Bessie has recently started playing a famous card game. In the game, there is only one deck of cards, consisting of $a$ "draw $0$" cards, $b$ "draw $1$" cards, $c$ "draw $2$" cards, and $5$ special cards. At the start of the game, all cards are in the randomly shuffled deck.
Bessie starts the game by drawing the top $5$ cards of the deck. She may then play "draw $x$" cards from the hand to draw the next $x$ cards from the top of the deck. Note that every card can only be played once, special cards cannot be played, and if Bessie uses a "draw $2$" card when there is only $1$ card remaining in the deck, then she simply draws that remaining card. Bessie wins if she draws all $5$ special cards.
Since Bessie is not very good at math problems, she wants you to find the probability that she wins, given that the deck is shuffled randomly over all $(a + b + c + 5)!$ possible orderings. It can be shown that this answer can always be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are coprime integers. Output $p \cdot q^{-1}$ modulo $998\,244\,353$.
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
Each test case contains three integers $a$, $b$, and $c$ ($0 \le a, b, c \le 2 \cdot 10^5$) – the number of draw $0$ cards, draw $1$ cards, and draw $2$ cards, respectively.
It is guaranteed that the sum of $a$ over all test cases does not exceed $2 \cdot 10^5$, the sum of $b$ over all test cases does not exceed $2 \cdot 10^5$, and the sum of $c$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, output a single integer — the probability that Bessie wins, modulo $998\,244\,353$.
Input
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
Each test case contains three integers $a$, $b$, and $c$ ($0 \le a, b, c \le 2 \cdot 10^5$) – the number of draw $0$ cards, draw $1$ cards, and draw $2$ cards, respectively.
It is guaranteed that the sum of $a$ over all test cases does not exceed $2 \cdot 10^5$, the sum of $b$ over all test cases does not exceed $2 \cdot 10^5$, and the sum of $c$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, output a single integer — the probability that Bessie wins, modulo $998\,244\,353$.
4
1 1 1
0 0 0
5 3 7
3366 1434 1234
903173463
1
35118742
398952013
Note
In the first case, we have $1$ of each type of "draw" card and $5$ special cards. There are $30\,720$ starting decks where Bessie will win by drawing the top $5$ cards and $40\,320$ starting decks in total. Thus, the probability of Bessie winning is $\frac{30\,720}{40\,320} = \frac{16}{21}$.
One example of a winning starting deck is, top to bottom,
- "Special",
- "Draw $1$",
- "Special",
- "Special",
- "Draw $0$",
- "Draw $2$",
- "Special",
- "Special".
One example of a losing starting deck is:
- "Special",
- "Draw $1$",
- "Special",
- "Special",
- "Draw $0$",
- "Special",
- "Special",
- "Draw $2$".