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Ex - Card Deck Score

Score : $600$ points

Problem Statement

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.

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.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$

$A_2$ $B_2$

$\vdots$

$A_N$ $B_N$

Output

Print the answer.

3 3
3 1
5 2
6 3

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 $819$.

3 2
1 1
5 2
25 1

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.

10 232657150901347497
139547946 28316250877914575
682142538 78223540024979445
110643588 74859962623690081
173455495 60713016476190629
271056265 85335723211047202
801329567 48049062628894325
864844366 54979173822804784
338794337 69587449430302156
737638908 15812229161735902
462149872 49993004923078537

39761306

Be sure to print the answer modulo $998244353$.