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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <cassert>
#include <cstdio>
#include <cmath>
#include <iostream>
#include <iomanip>
#include <sstream>
#include <vector>
#include <set>
#include <map>
#include <queue>
#include <numeric>
#include <algorithm>

using namespace std;
using lint = long long;
constexpr int MOD = 1000000007, INF = 1010101010;
constexpr lint LINF = 1LL << 60;

#ifdef _DEBUG
#include "../../library/library/debug_template.cpp"
#define DMP(...) dump(#__VA_ARGS__, __VA_ARGS__)
#else 
#define DMP(...) ((void)0)
#endif

struct init {
	init() {
		cin.tie(nullptr); ios::sync_with_stdio(false);
		cout << fixed << setprecision(10);
	}
} init_;



template<int Modulo = MOD>
struct Mint {

	lint val;
	constexpr Mint(lint v = 0) noexcept : val(v% Modulo) { if (val < 0) val += Modulo; }

	constexpr Mint& operator += (const Mint& r) noexcept {
		val += r.val;
		if (val >= Modulo) val -= Modulo;
		return *this;
	}
	constexpr Mint& operator -= (const Mint& r) noexcept {
		val -= r.val;
		if (val < 0) val += Modulo;
		return *this;
	}
	constexpr Mint& operator *= (const Mint& r) noexcept {
		val = val * r.val % Modulo;
		return *this;
	}
	constexpr Mint& operator /= (const Mint& r) noexcept {
		lint a = r.val, b = Modulo, u = 1, v = 0;
		while (b) {
			lint t = a / b;
			a -= t * b; swap(a, b);
			u -= t * v; swap(u, v);
		}
		val = val * u % Modulo;
		if (val < 0) val += Modulo;
		return *this;
	}

	constexpr Mint operator + (const Mint& r) const noexcept { return Mint(*this) += r; }
	constexpr Mint operator - (const Mint& r) const noexcept { return Mint(*this) -= r; }
	constexpr Mint operator * (const Mint& r) const noexcept { return Mint(*this) *= r; }
	constexpr Mint operator / (const Mint& r) const noexcept { return Mint(*this) /= r; }

	constexpr int getmod() { return Modulo; }
	constexpr Mint operator - () const noexcept { return val ? Modulo - val : 0; }

	constexpr bool operator == (const Mint& r) const noexcept { return val == r.val; }
	constexpr bool operator != (const Mint& r) const noexcept { return val != r.val; }

	friend ostream& operator << (ostream& os, const Mint<Modulo>& x) noexcept { return os << x.val; }
	friend istream& operator >> (istream& is, Mint<Modulo>& x) noexcept {
		lint tmp; is >> tmp;
		x = Mint(tmp);
		return is;
	}

	constexpr Mint pow(lint n) noexcept {
		Mint res{ 1 }, tmp{ val };
		while (n > 0) {
			if (n & 1) res *= tmp;
			tmp *= tmp;
			n >>= 1;
		}
		return res;
	}
};

using mint = Mint<>;

class Factorial {
private:
	vector<mint> fac, inv, finv;

	void build(int N) {
		fac[0] = fac[1] = 1; inv[1] = 1; finv[0] = finv[1] = 1;

		for (int i = 2; i < N; i++) {
			fac[i] = fac[i - 1] * i;
			inv[i] = -inv[MOD % i] * (MOD / i);
			finv[i] = finv[i - 1] * inv[i];
		}
	}

public:
	Factorial(int N = 1000000) : fac(N + 1), inv(N + 1), finv(N + 1) { build(N + 1); }

	mint Cmod(int n, int k) {
		if (n < k || k < 0) return 0LL;
		return fac[n] * (finv[k] * finv[n - k]);
	}

	mint Pmod(int n, int k) {
		if (n < k || k < 0) return 0LL;
		return fac[n] * finv[n - k];
	}
};

int main() {

	int N, M;
	cin >> N >> M;

	mint ans = 0;
	Factorial F;
	for (int i = 0; i <= N; i++) {
		int sgn = (i & 1) * -2 + 1;
		ans += F.Pmod(M, i) * F.Pmod(M - i, N - i) * F.Pmod(M - i, N - i) * F.Cmod(N, i) * sgn;
	}
	
	cout << ans << "\n";

	return 0;
}