Problem:
Score : 600 points

Problem Statement

There are N people, conveniently numbered 1 through N.
We want to divide them into some number of groups, under the following two conditions:

Every group contains between A and B people, inclusive.

Let F_i be the number of the groups containing exactly i people. Then, for all i, either F_i=0 or C≤F_i≤D holds.

Find the number of these ways to divide the people into groups.
Here, two ways to divide them into groups is considered different if and only if there exists two people such that they belong to the same group in exactly one of the two ways.
Since the number of these ways can be extremely large, print the count modulo 10^9+7.

Constraints

1≤N≤10^3

1≤A≤B≤N

1≤C≤D≤N

Input

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

N A B C D

Output

Print the number of ways to divide the people into groups under the conditions, modulo 10^9+7.

Sample Input 1

3 1 3 1 2

Sample Output 1

4

There are four ways to divide the people:

(1,2),(3)

(1,3),(2)

(2,3),(1)

(1,2,3)

The following way to divide the people does not count: (1),(2),(3). This is because it only satisfies the first condition and not the second.

Sample Input 2

7 2 3 1 3

Sample Output 2

105

The only ways to divide the people under the conditions are the ones where there are two groups of two people, and one group of three people.
There are 105 such ways.

Sample Input 3

1000 1 1000 1 1000

Sample Output 3

465231251

Sample Input 4

10 3 4 2 5

Sample Output 4

0

The answer can be 0.

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Convert;
using static System.Math;
//using Debug;
//using static System.Globalization.CultureInfo;
//using Edge = Pair<long, int>;
using System.Text;
class Program
{
    static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        Solve();
        Console.Out.Flush();
        //WriteLine(Solve());
    }
    static void Solve()
    {
        int num, a, b, c, d;
        Input.Make(out num, out a, out b, out c, out d);
        var comb = new Modulo.Combination((int)1e6 + 1);
        var dp = new ModInt[num + 10, num + 10];
        dp[0, 0] = 1;
        for(var i=1;i<=b-a+2;i++)
            for(var j=0;j<=num;j++)
            {
                var n = i + a-1;
                dp[i, j] = dp[i - 1, j];
                for (var k = c; k <= Min(j / n, d); k++)
                    dp[i, j] += dp[i - 1, j - n * k] * comb.Comb(num - j + n * k, n * k) * comb.fac(n*k) * ModInt.Pow(comb.facrev(n), k) * comb.facrev(k);
            }
        WriteLine(dp[b-a+1,num]);

    }
}

public class Input
{
    public static string read => ReadLine().Trim();
    public static int[] ar => read.Split(' ').Select(int.Parse).ToArray();
    public static int num => ToInt32(read);
    public static long[] arL => read.Split(' ').Select(long.Parse).ToArray();
    public static long numL => ToInt64(read);
    public static char[][] gred(int h)
        => Enumerable.Repeat(0, h).Select(_ => read.ToCharArray()).ToArray();
    public static int[] ar1D(int n)
        => Enumerable.Repeat(0, n).Select(_ => num).ToArray();
    public static long[] arL1D(int n)
        => Enumerable.Repeat(0, n).Select(_ => numL).ToArray();
    public static string[] strs(int n)
        => Enumerable.Repeat(0, n).Select(_ => read).ToArray();
    public static int[][] ar2D(int n)
        => Enumerable.Repeat(0, n).Select(_ => ar).ToArray();
    public static long[][] arL2D(int n)
        => Enumerable.Repeat(0, n).Select(_ => arL).ToArray();
    public static T getValue<T>(string g)
    {
        var t = typeof(T);
        if (t == typeof(int))
            return (T)(object)int.Parse(g);
        if (t == typeof(long))
            return (T)(object)long.Parse(g);
        if (t == typeof(string))
            return (T)(object)g;
        if (t == typeof(char))
            return (T)(object)char.Parse(g);
        if (t == typeof(double))
            return (T)(object)double.Parse(g);
        if (t == typeof(bool))
            return (T)(object)bool.Parse(g);
        return default(T);
    }
    public static string[] Make<T1, T2>(out T1 v1, out T2 v2)
    {
        var r = read.Split(' ');
        v1 = getValue<T1>(r[0]);
        v2 = getValue<T2>(r[1]);
        return r;
    }
    public static string[] Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3)
    {
        var r = Make(out v1, out v2);
        v3 = getValue<T3>(r[2]);
        return r;
    }
    public static string[] Make<T1,T2,T3,T4>(out T1 v1,out T2 v2,out T3 v3,out T4 v4)
    {
        var r = Make(out v1, out v2, out v3);
        v4 = getValue<T4>(r[3]);
        return r;
    }
    public static string[] Make<T1,T2,T3,T4,T5>(out T1 v1,out T2 v2,out T3 v3,out T4 v4,out T5 v5)
    {
        var r = Make(out v1, out v2, out v3, out v4);
        v5 = getValue<T5>(r[4]);
        return r;
    }
    public const long Inf = (long)1e18;
    public const double eps = 1e-6;
    public const string Alfa = "abcdefghijklmnopqrstuvwxyz";
    public const int MOD = 1000000007;
}

public class Modulo
{
    private static int Multiple(int num1, int num2)
        => (int)(BigMul(num1, num2) % Input.MOD);

    public static int Pow(int m, int n)
    {
        if (n == 0) return 1;
        if (n % 2 == 0) return Pow(Multiple(m, m), n >> 1);
        else return Multiple(Pow(Multiple(m, m), n >> 1), m);
    }

    public static int Div(int a, int b)
        => Multiple(a, Pow(b, Input.MOD - 2));

    public class Combination
    {
        private int[] _fac;
        private int[] _inv;
        private int[] _facrev;
        public Combination(int num)//階乗をメモ化
        {
            _fac = new int[num + 1];
            _facrev = new int[num + 1];
            _inv = new int[num + 1];
            _inv[1] = 1;
            _fac[0] = _fac[1] = 1;
            _facrev[0] = _facrev[1] = 1;
            for (var i = 2; i <= num; i++)
            {
                _fac[i] = Multiple(_fac[i - 1], i);
                _inv[i] = Input.MOD - Multiple(_inv[Input.MOD % i], Input.MOD / i);
                _facrev[i] = Multiple(_facrev[i - 1], _inv[i]);
            }
        }
        public int Comb(int n, int r)
        {
            if (n < r) return 0;
            if (n == r) return 1;
            var calc = _fac[n];
            calc = Multiple(calc, _facrev[r]);
            calc = Multiple(calc, _facrev[n-r]);
            return calc;
        }
        public int fac(int num)
            => _fac[num];
        public int facrev(int num)
            => _facrev[num];
    }

}

public struct ModInt
{
    public const long MOD = (int)1e9 + 7;
    public long num { get; set; }
    public ModInt(long n = 0) { num = n; }
    public override string ToString()
        => num.ToString();
    public static ModInt operator +(ModInt l, ModInt r)
    {
        l.num += r.num;
        if (l.num >= MOD) l.num -= MOD;
        return l;
    }
    public static ModInt operator -(ModInt l, ModInt r)
    {
        l.num -= r.num;
        if (l.num < 0) l.num += MOD;
        return l;
    }
    public static ModInt operator *(ModInt l, ModInt r)
        => new ModInt(l.num * r.num % MOD);
    public static ModInt operator /(ModInt l, ModInt r)
        => l * Pow(r, MOD - 2);
    public static implicit operator long(ModInt l) 
        => l.num;
    public static implicit operator ModInt(long n)
    {
        n %= MOD; if (n < 0) n += MOD;
        return new ModInt(n);
    }

    public static ModInt Pow(ModInt m,long n)
    {
        if (n == 0) return 1;
        if (n % 2 == 0) return Pow(m * m, n >> 1);
        else return Pow(m * m, n >> 1) * m;
    }
}