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 Java7 (OpenJDK 1.7.0) (judge verdict: Accepted)

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.NoSuchElementException;

public class Main {
  private final long MOD = (long) (1e9 + 7);

  public class Combination {
    private long[] fact;
    private long[] invFact;

    public Combination(int max) {
      long[] inv = new long[max + 1];
      fact = new long[max + 1];
      invFact = new long[max + 1];
      inv[1] = 1;
      for (int i = 2; i <= max; i++) inv[i] = inv[(int) (MOD % i)] * (MOD - MOD / i) % MOD;
      fact[0] = invFact[0] = 1;
      for (int i = 1; i <= max; i++) fact[i] = fact[i - 1] * i % MOD;
      for (int i = 1; i <= max; i++) invFact[i] = invFact[i - 1] * inv[i] % MOD;
    }

    public long get(int x, int y) {
      return fact[x] * invFact[y] % MOD * invFact[x - y] % MOD;
    }
  }

  private void solve(FastScanner in, PrintWriter out) {
    int N = in.nextInt();
    int A = in.nextInt();
    int B = in.nextInt();
    int C = in.nextInt();
    int D = in.nextInt();

    Combination combination = new Combination(N);

    long[][] dp = new long[N + 2][N + 2];
    dp[0][A] = 1;

    //dp[i][member] := i人を使ってmember人未満のチームで分ける方法
    for (int i = 0; i <= N; i++) {
      for (int member = A; member <= B; member++) {
        if (dp[i][member] <= 0) continue;
        dp[i][member + 1] += dp[i][member];
        dp[i][member + 1] %= MOD;

        long d = dp[i][member];
        for (int num = 1; num <= D; num++) {
          if (i + num * member > N) break;
          int remain = N - (i + (num - 1) * member);
          d *= combination.get(remain, member);
          d %= MOD;
          if (num < C) continue;
          long e = d * combination.invFact[num];
          e %= MOD;
          dp[i + num * member][member + 1] += e;
          dp[i + num * member][member + 1] %= MOD;
        }
      }
    }

    out.println(dp[N][B + 1]);

  }

  public static int[] getPrimes(int N) {
    ArrayList<Integer> primes = new ArrayList<>();
    boolean[] isComposite = new boolean[N + 1];
    primes.add(2);
    for (int i = 3; i <= N; i += 2)
      if (!isComposite[i]) {
        primes.add(i);
        for (int j = i * 2; j <= N; j += i) {
          isComposite[j] = true;
        }
      }
    int[] array = new int[primes.size()];
    for (int i = 0; i < primes.size(); i++) {
      array[i] = primes.get(i);
    }
    return array;
  }

  public static void main(String[] args) {
    FastScanner in = new FastScanner();
    PrintWriter out = new PrintWriter(System.out);
    new Main().solve(in, out);
    out.close();
  }

  private static class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int bufferLength = 0;

    private boolean hasNextByte() {
      if (ptr < bufferLength) {
        return true;
      } else {
        ptr = 0;
        try {
          bufferLength = in.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (bufferLength <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) return buffer[ptr++];
      else return -1;
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    private void skipUnprintable() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
    }

    boolean hasNext() {
      skipUnprintable();
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) throw new NoSuchElementException();
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    long nextLong() {
      if (!hasNext()) throw new NoSuchElementException();
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }

    double nextDouble() {
      return Double.parseDouble(next());
    }

    double[] nextDoubleArray(int n) {
      double[] array = new double[n];
      for (int i = 0; i < n; i++) {
        array[i] = nextDouble();
      }
      return array;
    }

    double[][] nextDoubleMap(int n, int m) {
      double[][] map = new double[n][];
      for (int i = 0; i < n; i++) {
        map[i] = nextDoubleArray(m);
      }
      return map;
    }

    public int nextInt() {
      return (int) nextLong();
    }

    public int[] nextIntArray(int n) {
      int[] array = new int[n];
      for (int i = 0; i < n; i++) array[i] = nextInt();
      return array;
    }

    public long[] nextLongArray(int n) {
      long[] array = new long[n];
      for (int i = 0; i < n; i++) array[i] = nextLong();
      return array;
    }

    public String[] nextStringArray(int n) {
      String[] array = new String[n];
      for (int i = 0; i < n; i++) array[i] = next();
      return array;
    }

    public char[][] nextCharMap(int n) {
      char[][] array = new char[n][];
      for (int i = 0; i < n; i++) array[i] = next().toCharArray();
      return array;
    }

    public int[][] nextIntMap(int n, int m) {
      int[][] map = new int[n][];
      for (int i = 0; i < n; i++) {
        map[i] = nextIntArray(m);
      }
      return map;
    }
  }
}