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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.util.*;

public class Main {

	private static Scanner sc;
	private static Printer pr;

	private static void solve() {
		int n = sc.nextInt();
		int a = sc.nextInt();
		int b = sc.nextInt();
		int c = sc.nextInt();
		int d = sc.nextInt();

		PC pc = new PC(n, MOD);

		long[] dp = new long[n + 1];
		{
			int min = Math.min(d, n / a);
			long[] tmp = new long[min + 1];
			tmp[0] = 1;
			for (int k = 1; k <= min; k++) {
				tmp[k] = tmp[k - 1] * pc.C(a * k, a) % MOD;
				tmp[k] *= pc.fact[k - 1];
				tmp[k] %= MOD;
				tmp[k] *= pc.ifact[k];
				tmp[k] %= MOD;
			}

			dp[0] = tmp[0];
			for (int k = c; k <= d && a * k <= n; k++) {
				dp[k * a] = tmp[k];
			}
		}

		for (int i = a + 1; i <= b; i++) {
			for (int j = n; j >= 0; j--) {
				int min = Math.min(d, j / i);
				long[] tmp = new long[min + 1];
				tmp[0] = 1;
				for (int k = 1; k <= min; k++) {
					tmp[k] = tmp[k - 1] * pc.C(j - i * (k - 1), i) % MOD;
					tmp[k] *= pc.fact[k - 1];
					tmp[k] %= MOD;
					tmp[k] *= pc.ifact[k];
					tmp[k] %= MOD;
				}

				for (int k = c; k <= d && i * k <= j; k++) {
					dp[j] += dp[j - i * k] * tmp[k] % MOD;
					dp[j] %= MOD;
				}
			}
		}

		pr.println(dp[n]);
	}

	private static final int MOD = 1_000_000_007;

	@SuppressWarnings("unused")
    private static class PC {
    	// MOD must be prime number.
    	int MOD;
    	// fact[i] : i! % MOD
    	long[] fact;
    	// ifact[i] : 1/i! % MOD
    	long[] ifact;

    	PC(int size, int MOD) {
    		// O(size)
    		// n=sizeまでのnCrを求める。
    		// nHrはn+r-1Crになってしまうので注意

    		this.MOD = MOD;

    		fact = new long[size + 1];
    		fact[0] = 1;
    		for (int i = 1; i <= size; i++) {
    			fact[i] = fact[i - 1] * i % MOD;
    		}

    		ifact = new long[size + 1];

    		int loop = MOD - 2;
    		long x = fact[size];
    		ifact[size] = 1;
    		while (loop > 0) {
    			if (loop % 2 == 1) {
    				ifact[size] = ifact[size] * x % MOD;
    			}
    			x = x * x % MOD;
    			loop /= 2;
    		}

    		for (int i = size - 1; i >= 0; i--) {
    			ifact[i] = ifact[i + 1] * (i + 1) % MOD;
    		}

    	}

    	// 組合せの数
    	int C(int n, int r) {
    		if (r > n) {
    			return 0;
    		}

    		return (int)(((fact[n] * ifact[n - r]) % MOD) * ifact[r] % MOD);
    	}

    	// 順列
    	int P(int n, int r) {
    		if (r > n) {
    			return 0;
    		}

    		return (int)((fact[n] * ifact[n -r]) % MOD);
    	}

    	// 重複組み合わせ
    	// 異なるn種のものから重複を許してr個を選ぶ場合の数
    	// 0個の種類もあり得る
    	int H(int n, int r) {
    		if (n == 0 && r == 0) {
    			return 1;
    		}

    		return C(n + r - 1, r);
    	}

    	// 組合せの数(nが大きいとき)
    	//   O(r)で求めることができる。rはsizeの大きさまで
    	int C2(long n, int r) {
    		long ret = ifact[r];
    		for (int i = 1; i <= r; i++) {
    			long tmp = (n - r + i) % MOD;
    			ret = (ret * tmp) % MOD;
    		}

    		return (int)ret;
    	}

    	// 第2種スターリング数
    	// n人をちょうどr個のグループに分ける(グループの区別はなし)
    	// グループの区別をする場合はr!S(n,r)。全射の場合の数と同義
    	// O(r log n)
    	int S(long n, int r) {
    		//全射の場合の数を包除原理を使って求めて、1/r!をかける。
    		long ret = 0;
    		for (int i = 1; i <= r; i++) {
    			long tmp = (r - i) % 2 == 0 ? 1 : -1;
    			tmp *= pow(i, n) * C(r, i) % MOD;
    			ret = (ret + tmp + MOD) % MOD;
    		}
    		ret = ret * ifact[r] % MOD;

    		return (int)ret;
    	}

    	long pow(int a, long n) {
    		long loop = n;
    		long ret = 1;
    		long x = a;
    		while (loop > 0) {
    			if (loop % 2 == 1) {
    				ret = ret * x % MOD;
    			}
    			x = x * x % MOD;
    			loop /= 2;
    		}

    		return ret;
    	}

    	// 組合せの数
    	// パスカルの三角形MODなし
		// 限界：n=66 : 66C33=7219428434016265740
    	private final static int LIMIT = 66;
    	private static int to;
    	private static long[][] cache;

    	static long CLong(int n, int r) {
    		if (r > n) {
    			return 0;
    		}

    		if (n > LIMIT) {
    			throw new IllegalArgumentException(Integer.toString(n));
    		}

    		if (cache == null) {
    			cache = new long[LIMIT + 1][];
    			cache[0] = new long[1];
    			cache[0][0] = 1;
    			to = 0;
    		}

    		if (cache[n] == null) {
    			for (int i = to + 1; i <= n; i++) {
    				cache[i] = new long[i + 1];
    				for (int j = 0; j <= i; j++) {
    					if (j == 0 || j == i) {
    						cache[i][j] = 1;
    					} else {
    						if (Long.MAX_VALUE - cache[i - 1][j - 1] < cache[i - 1][j]) {
    			    			throw new IllegalArgumentException("Overflow");
    						} else {
    							cache[i][j] = cache[i - 1][j - 1] + cache[i - 1][j];
    						}
    					}
    				}
    			}
    			to = n;
    		}

    		return cache[n][r];
    	}
    }

	// ---------------------------------------------------
	public static void main(String[] args) {
		sc = new Scanner(System.in);
		pr = new Printer(System.out);

		solve();

		pr.close();
		sc.close();
	}

	@SuppressWarnings("unused")
	private static class Scanner {
		BufferedReader br;

		Scanner (InputStream in) {
			br = new BufferedReader(new InputStreamReader(in));
		}

		private boolean isPrintable(int ch) {
			return ch >= '!' && ch <= '~';
		}

		private boolean isCRLF(int ch) {
			return ch == '\n' || ch == '\r' || ch == -1;
		}

		private int nextPrintable() {
			try {
				int ch;
				while (!isPrintable(ch = br.read())) {
					if (ch == -1) {
						throw new NoSuchElementException();
					}
				}

				return ch;
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		String next() {
			try {
				int ch = nextPrintable();
				StringBuilder sb = new StringBuilder();
				do {
					sb.appendCodePoint(ch);
				} while (isPrintable(ch = br.read()));

				return sb.toString();
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		int nextInt() {
			try {
				// parseInt from Integer.parseInt()
				boolean negative = false;
				int res = 0;
				int limit = -Integer.MAX_VALUE;
				int radix = 10;

				int fc = nextPrintable();
				if (fc < '0') {
					if (fc == '-') {
						negative = true;
						limit = Integer.MIN_VALUE;
					} else if (fc != '+') {
						throw new NumberFormatException();
					}
					fc = br.read();
				}
				int multmin = limit / radix;

				int ch = fc;
				do {
					int digit = ch - '0';
					if (digit < 0 || digit >= radix) {
						throw new NumberFormatException();
					}
					if (res < multmin) {
						throw new NumberFormatException();
					}
					res *= radix;
					if (res < limit + digit) {
						throw new NumberFormatException();
					}
					res -= digit;

				} while (isPrintable(ch = br.read()));

				return negative ? res : -res;
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		long nextLong() {
			try {
				// parseLong from Long.parseLong()
				boolean negative = false;
				long res = 0;
				long limit = -Long.MAX_VALUE;
				int radix = 10;

				int fc = nextPrintable();
				if (fc < '0') {
					if (fc == '-') {
						negative = true;
						limit = Long.MIN_VALUE;
					} else if (fc != '+') {
						throw new NumberFormatException();
					}
					fc = br.read();
				}
				long multmin = limit / radix;

				int ch = fc;
				do {
					int digit = ch - '0';
					if (digit < 0 || digit >= radix) {
						throw new NumberFormatException();
					}
					if (res < multmin) {
						throw new NumberFormatException();
					}
					res *= radix;
					if (res < limit + digit) {
						throw new NumberFormatException();
					}
					res -= digit;

				} while (isPrintable(ch = br.read()));

				return negative ? res : -res;
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		float nextFloat() {
			return Float.parseFloat(next());
		}

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

		String nextLine() {
			try {
				int ch;
				while (isCRLF(ch = br.read())) {
					if (ch == -1) {
						throw new NoSuchElementException();
					}
				}
				StringBuilder sb = new StringBuilder();
				do {
					sb.appendCodePoint(ch);
				} while (!isCRLF(ch = br.read()));

				return sb.toString();
			} catch (IOException e) {
				throw new NoSuchElementException();
			}
		}

		void close() {
			try {
				br.close();
			} catch (IOException e) {
//				throw new NoSuchElementException();
			}
		}
	}

	private static class Printer extends PrintWriter {
		Printer(PrintStream out) {
			super(out);
		}
	}
}