Problem:
Score : 500 points

Problem Statement

There are 2N squares arranged from left to right. You are given a string of length 2N representing the color of each of the squares.

The color of the i-th square from the left is black if the i-th character of S is B, and white if that character is W.

You will perform the following operation exactly N times: choose two distinct squares, then invert the colors of these squares and the squares between them. Here, to invert the color of a square is to make it white if it is black, and vice versa.

Throughout this process, you cannot choose the same square twice or more. That is, each square has to be chosen exactly once.

Find the number of ways to make all the squares white at the end of the process, modulo 10^9+7.

Two ways to make the squares white are considered different if and only if there exists i (1 \leq i \leq N) such that the pair of the squares chosen in the i-th operation is different.

Constraints

1 \leq N \leq 10^5

|S| = 2N

Each character of S is B or W.

Input

Input is given from Standard Input in the following format:

N
S

Output

Print the number of ways to make all the squares white at the end of the process, modulo 10^9+7. If there are no such ways, print 0.

Sample Input 1

2
BWWB

Sample Output 1

4

There are four ways to make all the squares white, as follows:

Choose Squares 1, 3 in the first operation, and choose Squares 2, 4 in the second operation.

Choose Squares 2, 4 in the first operation, and choose Squares 1, 3 in the second operation.

Choose Squares 1, 4 in the first operation, and choose Squares 2, 3 in the second operation.

Choose Squares 2, 3 in the first operation, and choose Squares 1, 4 in the second operation.

Sample Input 2

4
BWBBWWWB

Sample Output 2

288

Sample Input 3

5
WWWWWWWWWW

Sample Output 3

0

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

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;

class TEST{
	static void Main(){
		Sol mySol =new Sol();
		mySol.Solve();
	}
}

class Sol{
	public void Solve(){
		
		int N2 = N * 2;
		
		if(S[0] == 'W'){
			Console.WriteLine(0);
			return;
		}
		
		int[] LR = new int[N2];
		LR[0] = 0;
		for(int i=1;i<N2;i++){
			if(LR[i-1] == 0){
				if(S[i-1] == 'B' && S[i] == 'W') LR[i] = 0;
				if(S[i-1] == 'B' && S[i] == 'B') LR[i] = 1;
				if(S[i-1] == 'W' && S[i] == 'W') LR[i] = 1;
				if(S[i-1] == 'W' && S[i] == 'B') LR[i] = 0;
			} else {
				if(S[i-1] == 'B' && S[i] == 'W') LR[i] = 1;
				if(S[i-1] == 'B' && S[i] == 'B') LR[i] = 0;
				if(S[i-1] == 'W' && S[i] == 'W') LR[i] = 0;
				if(S[i-1] == 'W' && S[i] == 'B') LR[i] = 1;
			}
		}
		
		int[] cnt = new int[2];
		for(int i=0;i<N2;i++) cnt[LR[i]]++;
		if(cnt[0] != cnt[1]){
			Console.WriteLine(0);
			return;
		}
		
		//Console.WriteLine(String.Join("",LR));
		int dep = 0;
		int l = 0;
		int r = 0;
		mlong ans = 1;
		for(int i=0;i<N2;i++){
			if(LR[i] == 0) dep++;
			else dep--;
			if(dep < 0){
				Console.WriteLine(0);
				return;
			}
			if(LR[i] == 1){
				ans *= l;
				l--;
			}
			if(LR[i] == 0) l++;
			else  r++;
		}
		
		mlong p = 1;
		for(int i=0;i<N;i++) p *= (i + 1);
		
		Console.WriteLine(dep == 0 ? ans * p: 0);
		
		
		
		
		
	}
	int N;
	String S;
	public Sol(){
		N = ri();
		S = rs();
	}

	static String rs(){return Console.ReadLine();}
	static int ri(){return int.Parse(Console.ReadLine());}
	static long rl(){return long.Parse(Console.ReadLine());}
	static double rd(){return double.Parse(Console.ReadLine());}
	static String[] rsa(char sep=' '){return Console.ReadLine().Split(sep);}
	static int[] ria(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>int.Parse(e));}
	static long[] rla(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>long.Parse(e));}
	static double[] rda(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>double.Parse(e));}
}


struct mlong {
	
	const long mod = (long) 1e9 + 7;
	long V;
	
	public mlong(long _v = 0){
		V = _v;
	}
	
	public static mlong operator+(mlong a, mlong b){
		var v0 = a.V + b.V; if(v0 >= mod) v0 -= mod;
		return new mlong(v0);
	}
	public static mlong operator-(mlong a, mlong b){
		var v0 = mod + a.V - b.V; if(v0 >= mod) v0 -= mod;
		return new mlong(v0);
	}
	public static mlong operator-(mlong b){
		var v0 = mod - b.V; if(v0 >= mod) v0 -= mod;
		return new mlong(v0);
	}
	public static mlong operator*(mlong a, mlong b){
		var v0 = a.V * b.V; if(v0 >= mod) v0 %= mod;
		return new mlong(v0);
	}
	public static mlong operator/(mlong a, mlong b){
		var v0 = a.V * inv(b.V).V; if(v0 >= mod) v0 %= mod;
		return new mlong(v0);
	}
	
	
	public static mlong inv(long x){
		long a = 0, b = 0, c = 0;
		ExtGCD(x, mod, ref a, ref b, ref c);
		return (mlong)((a + mod) % mod);
	}
	
	public static void ExtGCD(long x, long y, ref long a, ref long b, ref long c){
		long r0 = x; long r1 = y;
		long a0 = 1; long a1 = 0;
		long b0 = 0; long b1 = 1;
		long q1, r2, a2, b2;
		while(r1 > 0){
			q1 = r0 / r1;
			r2 = r0 % r1;
			a2 = a0 - q1 * a1;
			b2 = b0 - q1 * b1;
			r0 = r1; r1 = r2;
			a0 = a1; a1 = a2;
			b0 = b1; b1 = b2;
		}
		c = r0;
		a = a0;
		b = b0;
	}
	
	public static mlong ModPow(mlong a, long k){
		if(k == 0) return (mlong) 1;
		if(a == 0) return (mlong) 0;
		mlong x = a;
		mlong ret = 1;
		while(k > 0){
			if(k % 2 == 1) ret *= x;
			x *= x;
			k >>= 1;
		}
		return ret;
	}
	
	
	public static bool operator == (mlong a, mlong b){
		 return a.Equals(b);
	}
	public static bool operator != (mlong a, mlong b){
		 return !(a == b);
	}
	
	public override bool Equals(System.Object obj){
		if(obj  ==  null) return false;
		mlong p  =  (mlong) obj;
		if((System.Object) p  ==  null) return false;
		return p.V  ==  V;
	}
	public override int GetHashCode(){
		return V.GetHashCode();
	}
	
	public static implicit operator mlong(long n){
		long v = n % mod; if(v < 0) v += mod;
		return new mlong(v);
	}
	public static implicit operator mlong(int n){
		long v = n % mod; if(v < 0) v += mod;
		return new mlong(v);
	}
	public static explicit operator long(mlong n){
		return n.V;
	}
	
	public override String ToString(){
		return V.ToString();
	}
	
	
}