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)

// ---------- begin ModularInteger ----------
use std::marker::*;
use std::ops::*;

trait Modulo {
    fn modulo() -> u32;
}

trait ConstantModulo {
    const MOD: u32;
}

impl<T> Modulo for T
where
    T: ConstantModulo,
{
    fn modulo() -> u32 {
        T::MOD
    }
}

struct ModularInteger<T>(u32, PhantomData<T>);

impl<T> Clone for ModularInteger<T> {
    fn clone(&self) -> Self {
        ModularInteger::new_unchecked(self.0)
    }
}

impl<T> Copy for ModularInteger<T> {}

impl<T: Modulo> Add for ModularInteger<T> {
    type Output = ModularInteger<T>;
    fn add(self, rhs: Self) -> Self::Output {
        let mut d = self.0 + rhs.0;
        if d >= T::modulo() {
            d -= T::modulo();
        }
        ModularInteger::new_unchecked(d)
    }
}

impl<T: Modulo> AddAssign for ModularInteger<T> {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl<T: Modulo> Sub for ModularInteger<T> {
    type Output = ModularInteger<T>;
    fn sub(self, rhs: Self) -> Self::Output {
        let mut d = T::modulo() + self.0 - rhs.0;
        if d >= T::modulo() {
            d -= T::modulo();
        }
        ModularInteger::new_unchecked(d)
    }
}

impl<T: Modulo> SubAssign for ModularInteger<T> {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl<T: Modulo> Mul for ModularInteger<T> {
    type Output = ModularInteger<T>;
    fn mul(self, rhs: Self) -> Self::Output {
        let v = self.0 as u64 * rhs.0 as u64 % T::modulo() as u64;
        ModularInteger::new_unchecked(v as u32)
    }
}

impl<T: Modulo> MulAssign for ModularInteger<T> {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl<T: Modulo> Neg for ModularInteger<T> {
    type Output = ModularInteger<T>;
    fn neg(self) -> Self::Output {
        if self.0 == 0 {
            Self::zero()
        } else {
            Self::new_unchecked(T::modulo() - self.0)
        }
    }
}

impl<T> std::fmt::Display for ModularInteger<T> {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl<T: Modulo> std::str::FromStr for ModularInteger<T> {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<u32>()?;
        Ok(ModularInteger::new(val))
    }
}

impl<T: Modulo> From<usize> for ModularInteger<T> {
    fn from(val: usize) -> ModularInteger<T> {
        ModularInteger::new_unchecked((val % T::modulo() as usize) as u32)
    }
}

impl<T: Modulo> From<i64> for ModularInteger<T> {
    fn from(val: i64) -> ModularInteger<T> {
        let m = T::modulo() as i64;
        ModularInteger::new((val % m + m) as u32)
    }
}

impl<T> ModularInteger<T> {
    fn new_unchecked(d: u32) -> Self {
        ModularInteger(d, PhantomData)
    }
    fn zero() -> Self {
        ModularInteger::new_unchecked(0)
    }
    fn one() -> Self {
        ModularInteger::new_unchecked(1)
    }
}

#[allow(dead_code)]
impl<T: Modulo> ModularInteger<T> {
    fn new(d: u32) -> Self {
        ModularInteger::new_unchecked(d % T::modulo())
    }
    fn pow(&self, mut n: u32) -> Self {
        let mut t = Self::one();
        let mut s = *self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    fn inv(&self) -> Self {
        assert!(self.0 != 0);
        self.pow(T::modulo() - 2)
    }
}

// ---------- end ModularInteger ----------
// ---------- begin Precalc ----------
#[allow(dead_code)]
struct Precalc<T> {
    inv: Vec<ModularInteger<T>>,
    fact: Vec<ModularInteger<T>>,
    ifact: Vec<ModularInteger<T>>,
}

#[allow(dead_code)]
impl<T: Modulo> Precalc<T> {
    pub fn new(n: usize) -> Precalc<T> {
        let mut inv = vec![ModularInteger::one(); n + 1];
        let mut fact = vec![ModularInteger::one(); n + 1];
        let mut ifact = vec![ModularInteger::one(); n + 1];
        for i in 2..(n + 1) {
            fact[i] = fact[i - 1] * ModularInteger::new_unchecked(i as u32);
        }
        ifact[n] = fact[n].inv();
        if n > 0 {
            inv[n] = ifact[n] * fact[n - 1];
        }
        for i in (1..n).rev() {
            ifact[i] = ifact[i + 1] * ModularInteger::new_unchecked((i + 1) as u32);
            inv[i] = ifact[i] * fact[i - 1];
        }
        Precalc {
            inv: inv,
            fact: fact,
            ifact: ifact,
        }
    }
    pub fn inv(&self, n: usize) -> ModularInteger<T> {
        assert!(n > 0);
        self.inv[n]
    }
    pub fn fact(&self, n: usize) -> ModularInteger<T> {
        self.fact[n]
    }
    pub fn ifact(&self, n: usize) -> ModularInteger<T> {
        self.ifact[n]
    }
    pub fn comb(&self, n: usize, k: usize) -> ModularInteger<T> {
        if k > n {
            return ModularInteger::zero();
        }
        self.fact[n] * self.ifact[k] * self.ifact[n - k]
    }
}
// ---------- end Precalc ----------

struct Mod;
impl ConstantModulo for Mod {
    const MOD: u32 = 1_000_000_007;
}

#[allow(dead_code)]
struct RuntimeMod;
static mut RUNTIME_MOD: u32 = 0;
impl Modulo for RuntimeMod {
    fn modulo() -> u32 {
        unsafe {
            RUNTIME_MOD
        }
    }
}

#[allow(dead_code)]
impl RuntimeMod {
    fn set_modulo(p: u32) {
        unsafe {
            RUNTIME_MOD = p;
        }
    }
}

type ModInt = ModularInteger<Mod>;

use proconio::*;
 
fn run() {
    input! {
        n: usize,
        m: usize,
    }
    let pc = Precalc::new(n + m);
    let mut ans = ModInt::zero();
    let mut sign = ModInt::one();
    for i in 0..=n {
        let mut way = pc.comb(n, i);
        way *= pc.fact(m) * pc.ifact(m - i);
        way *= (pc.fact(m - i) * pc.ifact(m - n)).pow(2);
        ans += sign * way;
        sign = -sign;
    }
    println!("{}", ans);
}
 
fn main() {
    run();
}