Problem:
Score : 800 points

Problem Statement

12:17 (UTC): The sample input 1 and 2 were swapped. The error is now fixed. We are very sorry for your inconvenience.

There are N children in AtCoder Kindergarten, conveniently numbered 1 through N. Mr. Evi will distribute C indistinguishable candies to the children.

If child i is given a candies, the child's happiness will become x_i^a, where x_i is the child's excitement level. The activity level of the kindergarten is the product of the happiness of all the N children.

For each possible way to distribute C candies to the children by giving zero or more candies to each child, calculate the activity level of the kindergarten. Then, calculate the sum over all possible way to distribute C candies. This sum can be seen as a function of the children's excitement levels x_1,..,x_N, thus we call it f(x_1,..,x_N).

You are given integers A_i,B_i (1≦i≦N). Find modulo 10^9+7.

Constraints

1≦N≦400

1≦C≦400

1≦A_i≦B_i≦400 (1≦i≦N)

Partial Score

400 points will be awarded for passing the test set satisfying A_i=B_i (1≦i≦N).

Input

The input is given from Standard Input in the following format:

N C
A_1 A_2 ... A_N
B_1 B_2 ... B_N

Output

Print the value of modulo 10^9+7.

Sample Input 1

2 3
1 1
1 1

Sample Output 1

4

This case is included in the test set for the partial score, since A_i=B_i.
We only have to consider the sum of the activity level of the kindergarten where the excitement level of both child 1 and child 2 are 1 (f(1,1)).

If child 1 is given 0 candy, and child 2 is given 3 candies, the activity level of the kindergarten is 1^0*1^3=1.

If child 1 is given 1 candy, and child 2 is given 2 candies, the activity level of the kindergarten is 1^1*1^2=1.

If child 1 is given 2 candies, and child 2 is given 1 candy, the activity level of the kindergarten is 1^2*1^1=1.

If child 1 is given 3 candies, and child 2 is given 0 candy, the activity level of the kindergarten is 1^3*1^0=1.

Thus, f(1,1)=1+1+1+1=4, and the sum over all f is also 4.

Sample Input 2

1 2
1
3

Sample Output 2

14

Since there is only one child, child 1's happiness itself will be the activity level of the kindergarten. Since the only possible way to distribute 2 candies is to give both candies to child 1, the activity level in this case will become the value of f.

When the excitement level of child 1 is 1, f(1)=1^2=1.

When the excitement level of child 1 is 2, f(2)=2^2=4.

When the excitement level of child 1 is 3, f(3)=3^2=9.

Thus, the answer is 1+4+9=14.

Sample Input 3

2 3
1 1
2 2

Sample Output 3

66

Since it can be seen that f(1,1)=4 , f(1,2)=15 , f(2,1)=15 , f(2,2)=32, the answer is 4+15+15+32=66.

Sample Input 4

4 8
3 1 4 1
3 1 4 1

Sample Output 4

421749

This case is included in the test set for the partial score.

Sample Input 5

3 100
7 6 5
9 9 9

Sample Output 5

139123417

Solution in Rust (1.15.1) (judge verdict: Accepted)

macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut iter = $ s . split_whitespace ( ) ; input_inner ! { iter , $ ( $ r ) * } } ; ( iter = $ iter : ident , $ ( $ r : tt ) * ) => { let s = { use std :: io :: Read ; let mut s = String :: new ( ) ; std :: io :: stdin ( ) . read_to_string ( & mut s ) . unwrap ( ) ; s } ; let mut $ iter = s . split_whitespace ( ) ; input_inner ! { $ iter , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { let s = { use std :: io :: Read ; let mut s = String :: new ( ) ; std :: io :: stdin ( ) . read_to_string ( & mut s ) . unwrap ( ) ; s } ; let mut iter = s . split_whitespace ( ) ; input_inner ! { iter , $ ( $ r ) * } } ; }
macro_rules ! input_inner { ( $ iter : expr ) => { } ; ( $ iter : expr , ) => { } ; ( $ iter : expr , mut $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let mut $ var = read_value ! ( $ iter , $ t ) ; input_inner ! { $ iter $ ( $ r ) * } } ; ( $ iter : expr , mut $ var : ident $ ( $ r : tt ) * ) => { input_inner ! { $ iter , mut $ var : usize $ ( $ r ) * } } ; ( $ iter : expr , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ iter , $ t ) ; input_inner ! { $ iter $ ( $ r ) * } } ; ( $ iter : expr , $ var : ident $ ( $ r : tt ) * ) => { input_inner ! { $ iter , $ var : usize $ ( $ r ) * } } ; }
macro_rules ! read_value { ( $ iter : expr , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ iter , $ t ) ) ,* ) } ; ( $ iter : expr , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ iter , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ iter : expr , { chars : $ base : expr } ) => { read_value ! ( $ iter , String ) . chars ( ) . map ( | c | ( c as u8 - $ base as u8 ) as usize ) . collect ::< Vec < usize >> ( ) } ; ( $ iter : expr , { char : $ base : expr } ) => { read_value ! ( $ iter , { chars : $ base } ) [ 0 ] } ; ( $ iter : expr , chars ) => { read_value ! ( $ iter , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ iter : expr , char ) => { read_value ! ( $ iter , chars ) [ 0 ] } ; ( $ iter : expr , usize1 ) => { read_value ! ( $ iter , usize ) - 1 } ; ( $ iter : expr , $ t : ty ) => { $ iter . next ( ) . unwrap ( ) . parse ::<$ t > ( ) . unwrap ( ) } ; }
const MOD: i64 = 1_000_000_007;
#[derive(Copy, Clone, Debug, Default, PartialEq, Eq, Hash)]
pub struct Modi64(pub i64);
impl Modi64 {
    #[inline]
    pub fn new(i: i64) -> Self {
        Modi64(Self::modulo(i, MOD))
    }
    #[inline]
    pub fn modulo(i: i64, m: i64) -> i64 {
        let mut x = i % m;
        if x < 0 {
            x += m;
        }
        x
    }
    pub fn pow(self, y: i64) -> Self {
        let mut y = Self::modulo(y, MOD - 1);
        let mut x = Self::new(1);
        let mut base = self;
        while y > 0 {
            if y & 1 == 1 {
                x *= base;
            }
            base *= base;
            y = y >> 1;
        }
        x
    }
    #[inline]
    pub fn inv(self) -> Self {
        let (mut x, mut s, mut t, mut u) = (1, self.0, MOD, 0);
        while t != 0 {
            let k = s / t;
            s -= k * t;
            std::mem::swap(&mut s, &mut t);
            x -= k * u;
            std::mem::swap(&mut x, &mut u);
        }
        Modi64::new(x)
    }
}
impl std::ops::Add for Modi64 {
    type Output = Self;
    #[inline]
    fn add(self, rhs: Self) -> Self::Output {
        Modi64((self.0 + rhs.0) % MOD)
    }
}
impl<'a> std::ops::Add<Modi64> for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn add(self, rhs: Modi64) -> Self::Output {
        *self + rhs
    }
}
impl<'a> std::ops::Add<&'a Modi64> for Modi64 {
    type Output = Modi64;
    #[inline]
    fn add(self, rhs: &'a Modi64) -> Self::Output {
        self + *rhs
    }
}
impl<'a> std::ops::Add for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn add(self, rhs: Self) -> Self::Output {
        *self + *rhs
    }
}
impl std::ops::AddAssign for Modi64 {
    #[inline]
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}
impl<'a> std::ops::AddAssign<&'a Modi64> for Modi64 {
    #[inline]
    fn add_assign(&mut self, rhs: &'a Modi64) {
        *self += *rhs;
    }
}
impl std::ops::Sub for Modi64 {
    type Output = Self;
    #[inline]
    fn sub(self, rhs: Self) -> Self::Output {
        Self::new(self.0 - rhs.0)
    }
}
impl<'a> std::ops::Sub<Modi64> for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn sub(self, rhs: Modi64) -> Self::Output {
        *self - rhs
    }
}
impl<'a> std::ops::Sub<&'a Modi64> for Modi64 {
    type Output = Modi64;
    #[inline]
    fn sub(self, rhs: &'a Modi64) -> Self::Output {
        self - *rhs
    }
}
impl<'a> std::ops::Sub for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn sub(self, rhs: Self) -> Self::Output {
        *self - *rhs
    }
}
impl std::ops::SubAssign for Modi64 {
    #[inline]
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}
impl<'a> std::ops::SubAssign<&'a Modi64> for Modi64 {
    #[inline]
    fn sub_assign(&mut self, rhs: &'a Modi64) {
        *self -= *rhs;
    }
}
impl std::ops::Mul for Modi64 {
    type Output = Self;
    #[inline]
    fn mul(self, rhs: Self) -> Self::Output {
        Modi64(self.0 * rhs.0 % MOD)
    }
}
impl<'a> std::ops::Mul<Modi64> for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn mul(self, rhs: Modi64) -> Self::Output {
        *self * rhs
    }
}
impl<'a> std::ops::Mul<&'a Modi64> for Modi64 {
    type Output = Modi64;
    #[inline]
    fn mul(self, rhs: &'a Modi64) -> Self::Output {
        self * *rhs
    }
}
impl<'a> std::ops::Mul for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn mul(self, rhs: Self) -> Self::Output {
        *self * *rhs
    }
}
impl std::ops::MulAssign for Modi64 {
    #[inline]
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}
impl<'a> std::ops::MulAssign<&'a Modi64> for Modi64 {
    #[inline]
    fn mul_assign(&mut self, rhs: &'a Modi64) {
        *self *= *rhs;
    }
}
impl std::ops::Div for Modi64 {
    type Output = Self;
    #[inline]
    fn div(self, rhs: Self) -> Self::Output {
        self * rhs.inv()
    }
}
impl<'a> std::ops::Div<Modi64> for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn div(self, rhs: Modi64) -> Self::Output {
        *self / rhs
    }
}
impl<'a> std::ops::Div<&'a Modi64> for Modi64 {
    type Output = Modi64;
    #[inline]
    fn div(self, rhs: &'a Modi64) -> Self::Output {
        self / *rhs
    }
}
impl<'a> std::ops::Div for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn div(self, rhs: Self) -> Self::Output {
        *self / *rhs
    }
}
impl std::ops::DivAssign for Modi64 {
    #[inline]
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}
impl<'a> std::ops::DivAssign<&'a Modi64> for Modi64 {
    #[inline]
    fn div_assign(&mut self, rhs: &'a Modi64) {
        *self /= *rhs;
    }
}
impl std::ops::Neg for Modi64 {
    type Output = Self;
    #[inline]
    fn neg(self) -> Self::Output {
        Self::new(-self.0)
    }
}
impl<'a> std::ops::Neg for &'a Modi64 {
    type Output = Modi64;
    #[inline]
    fn neg(self) -> Self::Output {
        -*self
    }
}
impl std::iter::Sum for Modi64 {
    #[inline]
    fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
        iter.fold(Self::new(0), std::ops::Add::add)
    }
}
impl<'a> std::iter::Sum<&'a Modi64> for Modi64 {
    #[inline]
    fn sum<I: Iterator<Item = &'a Self>>(iter: I) -> Self {
        iter.fold(Self::new(0), |x, &y| x + y)
    }
}
impl std::fmt::Display for Modi64 {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> Result<(), std::fmt::Error> {
        write!(f, "{}", self.0)
    }
}
fn main() {
    input! { n, c, a: [usize; n], b: [usize; n] };
    let mut x = vec![vec![Modi64(0); c + 1]; n];
    for i in 0..n {
        for j in a[i]..b[i] + 1 {
            let mut p = Modi64(1);
            for k in 0..c + 1 {
                x[i][k] += p;
                p *= Modi64(j as i64);
            }
        }
    }
    let mut dp = vec![vec![Modi64(0); c + 1]; n + 1];
    dp[0][c] = Modi64(1);
    for i in 0..n {
        for j in 0..c + 1 {
            for k in (0..j + 1).rev() {
                let d = dp[i][j];
                dp[i + 1][j - k] += d * x[i][k];
            }
        }
    }
    println!("{}", dp[n][0]);
}