Problem:
Score : 500 points

Problem Statement

Count the pairs of length-N sequences consisting of integers between 1 and M (inclusive), A_1, A_2, \cdots, A_{N} and B_1, B_2, \cdots, B_{N}, that satisfy all of the following conditions:

A_i \neq B_i, for every i such that 1\leq i\leq N.

A_i \neq A_j and B_i \neq B_j, for every (i, j) such that 1\leq i < j\leq N.

Since the count can be enormous, print it modulo (10^9+7).

Constraints

1\leq N \leq M \leq 5\times10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M

Output

Print the count modulo (10^9+7).

Sample Input 1

2 2

Sample Output 1

2

A_1=1,A_2=2,B_1=2,B_2=1 and A_1=2,A_2=1,B_1=1,B_2=2 satisfy the conditions.

Sample Input 2

2 3

Sample Output 2

18

Sample Input 3

141421 356237

Sample Output 3

881613484

Solution in Rust (1.42.0) (judge verdict: Accepted)

#[allow(unused_imports)]
use {
   proconio::{fastout, input, marker::*},
   std::cmp::*,
   std::collections::*,
};

#[fastout]
fn main() {
   input! {
      n: usize,
      m: usize
   }
   let c = Comb::new(m);
   let mut ans = c.p(m, n);
   let mut x = Mint::new(0);
   for i in 0..=n {
      if i % 2 == 1 {
         x -= c.c(n, i) * c.p(m - i, n - i);
      } else {
         x += c.c(n, i) * c.p(m - i, n - i);
      }
   }
   ans *= x;
   println!("{}", ans);
}

const MOD: i64 = 1000000007;
#[derive(Copy, Clone)]
pub struct Mint {
   x: i64,
}
impl Mint {
   fn new(x: i64) -> Self {
      Mint {
         x: (x % MOD + MOD) % MOD,
      }
   }
   fn pow(self, n: usize) -> Self {
      match n {
         0 => Mint::new(1),
         _ => {
            let mut a = self.pow(n >> 1);
            a *= a;
            if n & 1 == 1 {
               a *= self;
            }
            a
         }
      }
   }
   fn inv(self) -> Self { self.pow((MOD - 2) as usize) }
}
impl std::ops::Neg for Mint {
   type Output = Mint;
   fn neg(self) -> Self::Output { Self::new(-self.x) }
}
impl std::ops::AddAssign<Mint> for Mint {
   fn add_assign(&mut self, rhs: Self) {
      self.x += rhs.x;
      if self.x >= MOD {
         self.x -= MOD;
      }
   }
}
impl std::ops::AddAssign<i64> for Mint {
   fn add_assign(&mut self, rhs: i64) { *self += Mint::new(rhs); }
}
impl<T> std::ops::Add<T> for Mint
where
   Mint: std::ops::AddAssign<T>,
{
   type Output = Self;
   fn add(self, other: T) -> Self {
      let mut res = self;
      res += other;
      res
   }
}
impl std::ops::SubAssign<Mint> for Mint {
   fn sub_assign(&mut self, rhs: Self) {
      self.x += MOD - rhs.x;
      if self.x >= MOD {
         self.x -= MOD;
      }
   }
}
impl std::ops::SubAssign<i64> for Mint {
   fn sub_assign(&mut self, rhs: i64) { *self -= Mint::new(rhs); }
}
impl<T> std::ops::Sub<T> for Mint
where
   Mint: std::ops::SubAssign<T>,
{
   type Output = Self;
   fn sub(self, other: T) -> Self {
      let mut res = self;
      res -= other;
      res
   }
}
impl std::ops::MulAssign<Mint> for Mint {
   fn mul_assign(&mut self, rhs: Self) {
      self.x *= rhs.x;
      self.x %= MOD;
   }
}
impl std::ops::MulAssign<i64> for Mint {
   fn mul_assign(&mut self, rhs: i64) { *self *= Mint::new(rhs); }
}
impl<T> std::ops::Mul<T> for Mint
where
   Mint: std::ops::MulAssign<T>,
{
   type Output = Self;
   fn mul(self, other: T) -> Self {
      let mut res = self;
      res *= other;
      res
   }
}
impl std::ops::DivAssign<Mint> for Mint {
   fn div_assign(&mut self, rhs: Self) { *self *= rhs.inv(); }
}
impl std::ops::DivAssign<i64> for Mint {
   fn div_assign(&mut self, rhs: i64) { *self *= Mint::new(rhs).inv(); }
}
impl<T> std::ops::Div<T> for Mint
where
   Mint: std::ops::DivAssign<T>,
{
   type Output = Self;
   fn div(self, other: T) -> Self {
      let mut res = self;
      res /= other;
      res
   }
}
impl<T> From<T> for Mint
where
   i64: From<T>,
{
   fn from(x: T) -> Mint { Mint::new(i64::from(x)) }
}
impl std::fmt::Display for Mint {
   fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { write!(f, "{}", self.x) }
}
struct Comb {
   n: usize,
   fact: Vec<Mint>,
   ifact: Vec<Mint>,
}
impl Comb {
   fn new(n: usize) -> Self {
      let mut fact = vec![Mint::new(1); n + 1];
      let mut ifact = vec![Mint::new(1); n + 1];
      for i in 1..n + 1 {
         fact[i] = fact[i - 1] * i as i64;
      }
      ifact[n] = fact[n as usize].inv();
      for i in (1..n + 1).rev() {
         ifact[i - 1] = ifact[i] * i as i64;
      }
      Comb {
         n: n,
         fact: fact,
         ifact: ifact,
      }
   }
   fn c(&self, n: usize, r: usize) -> Mint {
      if r > n || n > self.n {
         Mint::new(0)
      } else {
         self.fact[n] * self.ifact[r] * self.ifact[n - r]
      }
   }
   fn p(&self, n: usize, r: usize) -> Mint {
      if r > n || n > self.n {
         Mint::new(0)
      } else {
         self.fact[n] * self.ifact[n - r]
      }
   }
}