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 Rust (1.15.1) (judge verdict: Accepted)

pub mod mod_int {
    use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};

    //we may need to change this value
    const MOD: i64 = 1e9 as i64 + 7;

    #[derive(Clone, Copy, Debug)]
    pub struct ModInt<T: Copy + Clone + Add + AddAssign + Mul + MulAssign + Sub + SubAssign>(pub T);

    #[warn(unused_macros)]
    macro_rules! mod_int_impl {
        ($($t:ty)*) => ($(
            impl Add<ModInt<$t>> for ModInt<$t> {
                type Output = ModInt<$t>;
                fn add(self, other: ModInt<$t>) -> ModInt<$t> {
                    self + other.0
                }
            }
            impl Add<$t> for ModInt<$t> {
                type Output = ModInt<$t>;
                fn add(self, rhs: $t) -> ModInt<$t> {
                    let mut tmp = rhs + self.0;
                    if tmp >= self::MOD as $t {
                        tmp = tmp - self::MOD as $t;
                    }
                    ModInt(tmp)
                }
            }
            impl Sub<ModInt<$t>> for ModInt<$t> {
                type Output = ModInt<$t>;
                fn sub(self, rhs: ModInt<$t>) -> ModInt<$t> {
                    self - rhs.0
                }
            }
            impl Sub<$t> for ModInt<$t> {
                type Output = ModInt<$t>;
                fn sub(self, rhs: $t) -> ModInt<$t> {
                    let rhs = if rhs >= self::MOD as $t { rhs % self::MOD as $t} else { rhs };
                    let value = if self.0 < rhs { self.0 + self::MOD as $t} else { self.0 };
                    ModInt(value - rhs)
                }
            }
            impl AddAssign<ModInt<$t>> for ModInt<$t> {
                fn add_assign(&mut self, rhs: ModInt<$t>) {
                    *self = *self + rhs;
                }
            }
            impl AddAssign<$t> for ModInt<$t> {
                fn add_assign(&mut self, rhs: $t) {
                    *self = *self + rhs;
                }
            }
            impl SubAssign<ModInt<$t>> for ModInt<$t> {
                fn sub_assign(&mut self, rhs: ModInt<$t>) {
                    *self = *self - rhs;
                }
            }

            impl SubAssign<$t> for ModInt<$t> {
                fn sub_assign(&mut self, rhs: $t) {
                    *self = *self - rhs;
                }
            }

            impl Mul<ModInt<$t>> for ModInt<$t> {
                type Output = ModInt<$t>;
                fn mul(self, rhs: ModInt<$t>) -> ModInt<$t> {
                    self * rhs.0
                }
            }
            impl Mul<$t> for ModInt<$t> {
                type Output = ModInt<$t>;
                fn mul(self, rhs: $t) -> ModInt<$t> {
                    ModInt(self.0 * rhs % self::MOD as $t)
                }
            }
            impl MulAssign<ModInt<$t>> for ModInt<$t> {
                fn mul_assign(&mut self, rhs: ModInt<$t>) {
                    *self = *self * rhs;
                }
            }

            impl MulAssign<$t> for ModInt<$t> {
                fn mul_assign(&mut self, rhs: $t) {
                    *self = *self * rhs;
                }
            }
            impl ModInt<$t> {
                pub fn new(x: $t) -> Self {
                    ModInt(x)
                }
                pub fn pow(self, e: usize) -> ModInt<$t> {
                    let mut result = ModInt::<$t>::new(1);
                    let mut cur = self;
                    let mut e = e;
                    while e > 0 {
                        if e & 1 == 1 {
                            result *= cur;
                        }
                        cur *= cur;
                        e >>= 1;
                    }
                    result
                }
            }
            )*)
    }
    mod_int_impl!(usize i32 i64 u32 u64);
}
type ModInt = mod_int::ModInt<usize>;
fn solve() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let s = sc.chars();

    if s[0] == 'W' {
        println!("0");
        return;
    }
    let mut open = 1;
    let mut result = ModInt::new(1);
    for &c in s.iter().skip(1) {
        if c == 'B' {
            if open % 2 == 0 {
                //open
                open += 1;
            } else {
                //close
                result *= open;
                open -= 1;
            }
        } else {
            //white
            if open % 2 == 0 {
                //close
                if open == 0 {
                    println!("0");
                    return;
                }
                result *= open;
                open -= 1;
            } else {
                open += 1;
            }
        }
    }
    if open != 0 {
        println!("0");
        return;
    }
    for i in 1..n + 1 {
        result *= i;
    }
    println!("{}", result.0);
}

fn main() {
    std::thread::Builder::new()
        .stack_size(64 * 1024 * 1024) // 64MB
        .spawn(|| solve())
        .unwrap()
        .join()
        .unwrap();
}

//snippet from kenkoooo
pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn read_line(&mut self) -> String {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b'\n' || b == b'\r')
            .take_while(|&b| b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }

    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}