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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest.ABC_172
{
    class E
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            long[] nm = ReadLongs();
            long n = nm[0];
            long m = nm[1];

            long mask = 1000000000 + 7;

            var calculator = new CaseCalculator(mask, m + 100);
            long[] evenRevSums = new long[n + 10];
            long[] oddRevSums = new long[n + 10];
            for (int i = 0; i < n + 10; i++)
            {
                if (i % 2 == 0)
                {
                    evenRevSums[i] = calculator.allInverses[i];
                }
                else
                {
                    oddRevSums[i] = calculator.allInverses[i];
                }

                if (i > 0)
                {
                    evenRevSums[i] += evenRevSums[i - 1];
                    evenRevSums[i] %= mask;
                    oddRevSums[i] += oddRevSums[i - 1];
                    oddRevSums[i] %= mask;
                }
            }

            long res = 0;
            for(int i = 0; i <= n; i++)
            {
                if (i % 2 == 0)
                {
                    res += (calculator.Combination(n, i) * calculator.Permutation(m-i, n - i)) % mask;
                    res %= mask;
                }
                else
                {
                    res += mask - (calculator.Combination(n, i) * calculator.Permutation(m-i, n - i)) % mask;
                    res %= mask;
                }
            }
            res *= calculator.Permutation(m, n);
            res %= mask;

            /*
            for(long i = n - (m - n); i <= n; i++)
            {
                long tmp = calculator.Permutation(n, n - i);
                long selects = calculator.allPermutations[i];
                if (i % 2 == 0)
                {
                    selects *= (mask + evenRevSums[i] - oddRevSums[i]) % mask;

                }
                else
                {
                    selects *= (mask + oddRevSums[i] - evenRevSums[i]) % mask;
                }
                selects %= mask;
                tmp *= selects;
                tmp %= mask;
                res += tmp;
            }

            res *= calculator.Permutation(m, n);
            res %= mask;
            */
            WriteLine(res);
        }

        class CaseCalculator
        {
            long mask;
            public long[] allPermutations;
            public long[] allInverses;

            public CaseCalculator(long mask, long permutationCnt)
            {
                this.mask = mask;
                allPermutations = AllPermutations(permutationCnt);
                allInverses = AllInverses(permutationCnt);
            }

            public long Combination(long n, long m)
            {
                if (n < m) return 0;

                if (n - m < m) m = n - m;

                return Multi(allPermutations[n], allInverses[n - m], allInverses[m]);
            }

            public long Permutation(long n, long m)
            {
                if (n < m) return 0;

                return Multi(allPermutations[n], allInverses[n - m]);
            }

            long[] AllPermutations(long n)
            {
                var perms = new long[n + 1];
                perms[0] = 1;
                for (int i = 1; i <= n; i++)
                {
                    perms[i] = Multi(perms[i - 1], i);
                }
                return perms;
            }

            long[] AllInverses(long n)
            {
                var inverses = new long[n + 1];
                inverses[1] = 1;
                inverses[1] = 1;
                var permInverses = new long[n + 1];
                permInverses[0] = 1;
                permInverses[1] = 1;
                for (int i = 2; i <= n; i++)
                {
                    inverses[i] = mask - inverses[mask % i] * (mask / i) % mask;
                    permInverses[i] = Multi(permInverses[i - 1], inverses[i]);
                }
                return permInverses;
            }

            public long Multi(params long[] vals)
            {
                if (vals.Length == 0)
                {
                    return 0;
                }

                long val = vals[0] % mask;
                for (int i = 1; i < vals.Length; i++)
                {
                    val *= (vals[i] % mask);
                    val %= mask;
                }
                return val;
            }

            public long Inverse(long val)
            {
                long a = mask;
                long b = val % mask;
                long x = 1;
                long x1 = 0;
                long y = 0;
                long y1 = 1;
                while (b > 0)
                {
                    long q = a / b;
                    long r = a % b;
                    long x2 = (mask + x - (q * x1) % mask) % mask;
                    long y2 = (mask + y - (q * y1) % mask) % mask;
                    a = b;
                    b = r;
                    x = x1;
                    x1 = x2;
                    y = y1;
                    y1 = y2;
                }
                return y;
            }

            long DeprecatedInverse(long val)
            {
                return Pow(val, mask - 2);
            }

            long Pow(long a, long exp)
            {
                if (exp == 0) return 1;
                else if (exp == 1) return a % mask;
                else
                {
                    long halfMod = Pow(a, exp / 2);
                    long nextMod = Multi(halfMod, halfMod);
                    if (exp % 2 == 0)
                    {
                        return nextMod;
                    }
                    else
                    {
                        return Multi(nextMod, a);
                    }
                }
            }
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}