9568: ABC241 —— Ex - Card Deck Score
[Creator : ]
Description
There are some cards. Each card has one of $N$ integers written on it. Specifically, there are $B_i$ cards with $A_i$ written on them.
Next, for a combination of $M$ cards chosen out of these $(B_1+B_2\cdots +B_N)$ cards, we define the score of the combination by the product of the integers written on the $M$ cards.
Supposed that cards with the same integer written on them are indistinguishable, find the sum, modulo $998244353$, of the scores over all possible combinations of $M$ cards.
Next, for a combination of $M$ cards chosen out of these $(B_1+B_2\cdots +B_N)$ cards, we define the score of the combination by the product of the integers written on the $M$ cards.
Supposed that cards with the same integer written on them are indistinguishable, find the sum, modulo $998244353$, of the scores over all possible combinations of $M$ cards.
Input
Input is given from Standard Input in the following format:
```
$N$ $M$
$A_1$ $B_1$
$A_2$ $B_2$
$\vdots$
$A_N$ $B_N$
```
```
$N$ $M$
$A_1$ $B_1$
$A_2$ $B_2$
$\vdots$
$A_N$ $B_N$
```
Output
Print the answer.
Constraints
- $1 \leq N \leq 16$
- $1 \leq M \leq 10^{18}$
- $1 \leq A_i < 998244353$
- $1 \leq B_i \leq 10^{17}$
- If $i\neq j$, then $A_i \neq A_j$.
- $M\leq B_1+B_2+\cdots B_N$
- All values in input are integers.
- $1 \leq M \leq 10^{18}$
- $1 \leq A_i < 998244353$
- $1 \leq B_i \leq 10^{17}$
- If $i\neq j$, then $A_i \neq A_j$.
- $M\leq B_1+B_2+\cdots B_N$
- All values in input are integers.
Sample 1 Input
3 3
3 1
5 2
6 3
Sample 1 Output
819
There are 6 possible combinations of 3 cards.
- A combination of 1 card with 3 written on it, and 2 cards with 5 written on them.
- A combination of 1 card with 3 written on it, 1 card with 5 written on it, and 1 card with 6 written on it.
- A combination of 1 card with 3 written on it, and 2 cards with 6 written on them.
- A combination of 2 cards with 5 written on them, and 1 card with 6 written on it.
- A combination of 1 card with 5 written on it, and 2 cards with 6 written on them.
- A combination of 3 cards with 6 written on them.
The scores are 75, 90, 108, 150, 180, and 216, respectively, for a sum of 819.
Sample 2 Input
3 2
1 1
5 2
25 1
Sample 2 Output
180
"A combination of a card with 1 and another card with 25" and "a combination of two cards with 5 written on them" have the same score of 25, but they are considered to be different combinations.
Sample 3 Input
10 232657150901347497
139547946 28316250877914575
682142538 78223540024979445
110643588 74859962623690081
173455495 60713016476190629
271056265 85335723211047202
801329567 48049062628894325
864844366 54979173822804784
338794337 69587449430302156
737638908 15812229161735902
462149872 49993004923078537
Sample 3 Output
39761306