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H - Random Kth Max

Score : $600$ points

Problem Statement

We have $N$ continuous random variables $X_1,X_2,\dots,X_N$. $X_i$ has a continuous uniform distribution over the interval $\lbrack L_i, R_i \rbrack$.
Let $E$ be the expected value of the $K$-th greatest value among the $N$ random variables. Print $E \bmod {998244353}$ as specified in Notes.

Notes

In this problem, we can prove that $E$ is always a rational number. Additionally, the Constraints of this problem guarantee that, when $E$ is represented as an irreducible fraction $\frac{y}{x}$, $x$ is indivisible by $998244353$.
Here, there uniquely exists an integer $z$ between $0$ and $998244352$ such that $xz \equiv y \pmod{998244353}$. Print this $z$ as the value $E \bmod {998244353}$.

Constraints

• $1 \leq N \leq 50$
• $1 \leq K \leq N$
• $0 \leq L_i \lt R_i \leq 100$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$L_1$ $R_1$

$L_2$ $R_2$

$\vdots$

$L_N$ $R_N$

Output

Print $E \bmod {998244353}$.

1 1
0 2

1

The answer is the expected value of the random variable with a continuous uniform distribution over the interval $\lbrack 0, 2 \rbrack$. Thus, we should print $1$.

2 2
0 2
1 3

707089751

The answer represented as a rational number is $\frac{23}{24}$. We have $707089751 \times 24 \equiv 23 \pmod{998244353}$, so we should print $707089751$.

10 5
35 48
44 64
47 59
39 97
36 37
4 91
38 82
20 84
38 50
39 69

810056397