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: Time Limit Exceeded)

use std::cmp::*;
use std::io::*;
use std::ops::*;
use utils::*;

pub fn main() {
    let i = stdin();
    let mut o = Vec::new();
    run(i.lock(), &mut o);
    stdout().write_all(&o).unwrap();
}

pub fn run<R: BufRead, W: Write>(i: R, o: &mut W) {
    let mut i = CpReader::new(i);
    let n = i.read::<usize>();
    let s = i.read_line();

    writeln!(o, "{}", solve(n, &s)).unwrap();
}
fn solve(n: usize, s: &[u8]) -> usize {
    let mut cs = vec![Mod::new(0); n + 2];
    let mut cs_next = vec![Mod::new(0); n + 2];
    cs[0] += 1;

    let mut end = 1;
    for &s in s {
        for i in 0..cs_next.len() {
            cs_next[i] = Mod::new(0);
        }

        if s == b'W' {
            if end % 2 == 0 {
                end += 1;
            }
            end = min(end, n + 1);
            for i in 0..end {
                if i % 2 == 0 {
                    if i != 0 {
                        cs_next[i] += cs[i - 1];
                    }
                } else {
                    cs_next[i] += cs[i + 1] * (i + 1);
                }
            }
        } else {
            if end % 2 != 0 {
                end += 1;
            }
            end = min(end, n + 1);
            for i in 0..end {
                if i % 2 == 0 {
                    cs_next[i] += cs[i + 1] * (i + 1);
                } else {
                    cs_next[i] += cs[i - 1];
                }
            }
        }
        std::mem::swap(&mut cs, &mut cs_next);
    }
    let mut c = Mod::new(1);
    for i in 0..n {
        c *= i + 1;
    }
    (c * cs[0]).0
}

pub mod utils {
    use super::*;

    pub struct CpReader<R: BufRead> {
        r: R,
        b: Vec<u8>,
    }
    impl<R: BufRead> CpReader<R> {
        pub fn new(r: R) -> Self {
            CpReader {
                r: r,
                b: Vec::new(),
            }
        }
        pub fn read_word(&mut self) -> &[u8] {
            self.b.clear();
            let mut consume = 0;
            loop {
                self.r.consume(consume);
                let b = self.r.fill_buf().unwrap();
                assert!(b.len() != 0);
                if let Some(p) = b.iter().position(|&x| x == b' ' || x == b'\n') {
                    self.b.extend_from_slice(&b[..p]);
                    consume = p + 1;
                    break;
                }
                self.b.extend_from_slice(b);
                consume = b.len();
            }
            self.r.consume(consume);
            &self.b
        }
        pub fn read_word_str(&mut self) -> &str {
            unsafe { std::str::from_utf8_unchecked(self.read_word()) }
        }

        pub fn read_line(&mut self) -> &[u8] {
            self.b.clear();
            self.r.read_until(b'\n', &mut self.b).unwrap();
            if self.b.last() == Some(&b'\n') {
                &self.b[..self.b.len() - 1]
            } else {
                &self.b
            }
        }
        pub fn read_line_str(&mut self) -> &str {
            unsafe { std::str::from_utf8_unchecked(self.read_line()) }
        }

        pub fn read<T: CpIn>(&mut self) -> T {
            T::read_from(self)
        }

        pub fn read_vec<T: CpIn>(&mut self, n: usize) -> Vec<T> {
            (0..n).map(|_| self.read()).collect()
        }
        pub fn read_iter<'a, T: CpIn>(&'a mut self, n: usize) -> CpIter<'a, R, T> {
            CpIter {
                r: self,
                n: n,
                _pd: Default::default(),
            }
        }
    }
    pub struct CpIter<'a, R: BufRead + 'a, T> {
        r: &'a mut CpReader<R>,
        n: usize,
        _pd: std::marker::PhantomData<fn() -> T>,
    }
    impl<'a, R: BufRead, T: CpIn> Iterator for CpIter<'a, R, T> {
        type Item = T;
        fn next(&mut self) -> Option<T> {
            if self.n == 0 {
                None
            } else {
                self.n -= 1;
                Some(self.r.read())
            }
        }
    }

    pub trait CpIn {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self;
    }

    impl CpIn for u64 {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            read_u64_fast(&mut r.r)
        }
    }
    impl CpIn for i64 {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            read_i64_fast(&mut r.r)
        }
    }
    impl CpIn for char {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            let b = r.r.fill_buf().unwrap()[0] as char;
            r.r.consume(1);
            let s = r.r.fill_buf().unwrap()[0];
            assert!(s == b' ' || s == b'\n');
            r.r.consume(1);
            b
        }
    }

    macro_rules! cpin_tuple {
        ($($t:ident),*) => {
            impl<$($t: CpIn),*> CpIn for ($($t),*) {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    ($($t::read_from(r)),*)
                }
            }
        };
    }

    macro_rules! cpin_cast {
        ($t_self:ty, $t_read:ty) => {
            impl CpIn for $t_self {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    <$t_read>::read_from(r) as $t_self
                }
            }
        };
    }
    macro_rules! cpin_parse {
        ($t:ty) => {
            impl CpIn for $t {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    r.read_word_str().parse().unwrap()
                }
            }
        };
    }
    cpin_cast!(usize, u64);
    cpin_cast!(u32, u64);
    cpin_cast!(u16, u64);
    cpin_cast!(i32, i64);
    cpin_cast!(i16, i64);
    cpin_cast!(i8, i64);

    cpin_parse!(f64);

    cpin_tuple!(T1, T2);
    cpin_tuple!(T1, T2, T3);
    cpin_tuple!(T1, T2, T3, T4);
    cpin_tuple!(T1, T2, T3, T4, T5);

    fn read_u64_fast<R: BufRead>(r: &mut R) -> u64 {
        let mut value = 0;
        loop {
            let mut idx = 0;
            let buf_len;
            {
                let buf = r.fill_buf().unwrap();
                buf_len = buf.len();
                assert!(buf_len != 0);
                while idx < buf_len && b'0' <= buf[idx] && buf[idx] <= b'9' {
                    value = value * 10 + (buf[idx] - b'0') as u64;
                    idx += 1;
                }
            }
            if idx < buf_len {
                r.consume(idx + 1);
                return value;
            }
            r.consume(idx);
        }
    }
    fn read_i64_fast<R: BufRead>(r: &mut R) -> i64 {
        let (consume, sign) = match r.fill_buf().unwrap()[0] {
            b'+' => (true, 1),
            b'-' => (true, -1),
            _ => (false, 1),
        };
        if consume {
            r.consume(1);
        }
        read_u64_fast(r) as i64 * sign
    }

    #[derive(Clone, Copy)]
    pub struct Mod(pub usize);

    use std::fmt;
    impl fmt::Debug for Mod {
        fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
            write!(f, "{:?}", self.0)
        }
    }

    const MOD_BASE: usize = 1000000007;
    const MOD_INV_POW: usize = MOD_BASE - 2;

    impl Mod {
        pub fn new(n: usize) -> Self {
            Mod(n % MOD_BASE)
        }
        pub fn pow(self, mut exp: usize) -> Self {
            let mut b = self;
            let mut r = Mod(1);
            while exp != 0 {
                if exp % 2 == 1 {
                    r *= b;
                }
                exp /= 2;
                b = b * b;
            }
            r
        }
        pub fn inverse_element(self) -> Self {
            self.pow(MOD_INV_POW)
        }

        pub fn comb(n: usize, m: usize) -> Mod {
            let mut v0 = Mod::new(1);
            let mut v1 = Mod::new(1);
            for i in 0..min(m, n - m) {
                v0 *= i + 1;
                v1 *= n - i;
            }
            v1 / v0
        }
    }

    macro_rules! mod_op {
        (($ot:ident, $f:ident), ($ot_a:ident, $f_a:ident), $($rhs:ty : $e:expr,)*) => {
            $(
            impl $ot<$rhs> for Mod {
                type Output = Self;
                fn $f(self, rhs: $rhs) -> Self {
                    ($e)(self, rhs)
                }
            }
            impl $ot_a<$rhs> for Mod {
                fn $f_a(&mut self, rhs: $rhs) {
                    *self = ($e)(*self, rhs)
                }
            }
            )*
        }
    }

    mod_op!(
        (Add, add),
        (AddAssign, add_assign),
        Mod : |l: Mod, r: Mod| Mod::new(l.0 + r.0),
        usize : |l: Mod, r: usize| l + Mod::new(r),
    );

    mod_op!(
        (Mul, mul),
        (MulAssign, mul_assign),
        Mod : |l: Mod, r: Mod| Mod::new(l.0 * r.0),
        usize : |l: Mod, r: usize| l * Mod::new(r),
    );

    mod_op!(
        (Sub, sub),
        (SubAssign, sub_assign),
        Mod : |l: Mod, r: Mod| Mod::new(MOD_BASE + l.0 - r.0),
        usize : |l: Mod, r: usize| l - Mod::new(r),
    );

    mod_op!(
        (Div, div),
        (DivAssign, div_assign),
        Mod : |l: Mod, r: Mod| l * r.inverse_element(),
        usize : |l: Mod, r: usize| l / Mod::new(r),
    );
}