Problem:
Score : 800 points

Problem Statement

There are N children, numbered 1,2,\ldots,N.
In the next K days, we will give them some cookies.
In the i-th day, we will choose a_i children among the N with equal probability, and give one cookie to each child chosen. (We make these K choices independently.)

Let us define the happiness of the children as c_1 \times c_2 \times \ldots \times c_N, where c_i is the number of cookies received by Child i in the K days.
Find the expected happiness multiplied by \binom{N}{a_1} \times \binom{N}{a_2} \times \ldots \times \binom{N}{a_K} (we can show that this value is an integer), modulo (10^{9}+7).

Notes

\binom{n}{k} denotes the number of possible choices of k objects out of given distinct n objects, disregarding order.

Constraints

1 \leq N \leq 1000

1 \leq K \leq 20

1 \leq a_i \leq N

Input

Input is given from Standard Input in the following format:

N K
a_1 a_2 \ldots a_K

Output

Print the answer.

Sample Input 1

3 2
3 2

Sample Output 1

12

On the first day, all the children 1, 2, and 3 receive one cookie.

On the second day, two out of the three children 1, 2, and 3 receive one cookie.

Their happiness is 4 in any case, so the expected happiness is 4. Print this value multiplied by \binom{3}{3} \times \binom{3}{2}, which is 12.

Sample Input 2

856 16
399 263 665 432 206 61 784 548 422 313 848 478 827 26 398 63

Sample Output 2

337587117

Find the expected value multiplied by \binom{N}{a_1} \times \binom{N}{a_2} \times \ldots \times \binom{N}{a_K}, modulo (10^9+7).

Solution in Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(non_snake_case, unused)]

use std::cmp::*;
use std::collections::*;
use std::ops::*;

macro_rules! eprint {
	($($t:tt)*) => {{
		use ::std::io::Write;
		let _ = write!(::std::io::stderr(), $($t)*);
	}};
}
macro_rules! eprintln {
	() => { eprintln!(""); };
	($($t:tt)*) => {{
		use ::std::io::Write;
		let _ = writeln!(::std::io::stderr(), $($t)*);
	}};
}
macro_rules! dbg {
	($v:expr) => {{
		let val = $v;
		eprintln!("[{}:{}] {} = {:?}", file!(), line!(), stringify!($v), val);
		val
	}}
}

macro_rules! mat {
	($($e:expr),*) => { Vec::from(vec![$($e),*]) };
	($($e:expr,)*) => { Vec::from(vec![$($e),*]) };
	($e:expr; $d:expr) => { Vec::from(vec![$e; $d]) };
	($e:expr; $d:expr $(; $ds:expr)+) => { Vec::from(vec![mat![$e $(; $ds)*]; $d]) };
}

macro_rules! ok {
	($a:ident$([$i:expr])*.$f:ident()$(@$t:ident)*) => {
		$a$([$i])*.$f($($t),*)
	};
	($a:ident$([$i:expr])*.$f:ident($e:expr$(,$es:expr)*)$(@$t:ident)*) => { {
		let t = $e;
		ok!($a$([$i])*.$f($($es),*)$(@$t)*@t)
	} };
}

pub fn readln() -> String {
	let mut line = String::new();
	::std::io::stdin().read_line(&mut line).unwrap_or_else(|e| panic!("{}", e));
	line
}

macro_rules! read {
	($($t:tt),*; $n:expr) => {{
		let stdin = ::std::io::stdin();
		let ret = ::std::io::BufRead::lines(stdin.lock()).take($n).map(|line| {
			let line = line.unwrap();
			let mut it = line.split_whitespace();
			_read!(it; $($t),*)
		}).collect::<Vec<_>>();
		ret
	}};
	($($t:tt),*) => {{
		let line = readln();
		let mut it = line.split_whitespace();
		_read!(it; $($t),*)
	}};
}

macro_rules! _read {
	($it:ident; [char]) => {
		_read!($it; String).chars().collect::<Vec<_>>()
	};
	($it:ident; [u8]) => {
		Vec::from(_read!($it; String).into_bytes())
	};
	($it:ident; usize1) => {
		$it.next().unwrap_or_else(|| panic!("input mismatch")).parse::<usize>().unwrap_or_else(|e| panic!("{}", e)) - 1
	};
	($it:ident; [usize1]) => {
		$it.map(|s| s.parse::<usize>().unwrap_or_else(|e| panic!("{}", e)) - 1).collect::<Vec<_>>()
	};
	($it:ident; [$t:ty]) => {
		$it.map(|s| s.parse::<$t>().unwrap_or_else(|e| panic!("{}", e))).collect::<Vec<_>>()
	};
	($it:ident; $t:ty) => {
		$it.next().unwrap_or_else(|| panic!("input mismatch")).parse::<$t>().unwrap_or_else(|e| panic!("{}", e))
	};
	($it:ident; $($t:tt),+) => {
		($(_read!($it; $t)),*)
	};
}

pub fn main() {
	let _ = ::std::thread::Builder::new().name("run".to_string()).stack_size(32 * 1024 * 1024).spawn(run).unwrap().join();
}

const MOD: usize = 1_000_000_007;
const INF: i64 = std::i64::MAX;

pub trait SetMinMax {
	fn setmin(&mut self, v: Self) -> bool;
	fn setmax(&mut self, v: Self) -> bool;
}
impl<T> SetMinMax for T where T: PartialOrd {
	fn setmin(&mut self, v: T) -> bool {
		*self > v && { *self = v; true }
	}
	fn setmax(&mut self, v: T) -> bool {
		*self < v && { *self = v; true }
	}
}

use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

type Num = usize;

#[derive(Clone, Copy, Debug)]
pub struct ModInt<T: Copy + Clone>(pub T);

impl Add<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self + rhs.0
    }
}

impl Add<Num> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn add(self, rhs: Num) -> ModInt<Num> {
        let mut t = rhs + self.0;
        if t >= MOD {
            t = t - MOD;
        }
        ModInt(t)
    }
}

impl Sub<Num> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn sub(self, rhs: Num) -> ModInt<Num> {
        let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
        let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
        ModInt(value - rhs)
    }
}

impl Sub<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self - rhs.0
    }
}

impl AddAssign<Num> for ModInt<Num> {
    fn add_assign(&mut self, other: Num) {
        *self = *self + other;
    }
}
impl AddAssign<ModInt<Num>> for ModInt<Num> {
    fn add_assign(&mut self, other: ModInt<Num>) {
        *self = *self + other;
    }
}

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

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

impl Div<Num> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn div(self, rhs: Num) -> ModInt<Num> {
        self * ModInt(rhs).pow(MOD - 2)
    }
}

impl Div<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn div(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self / rhs.0
    }
}

impl DivAssign<Num> for ModInt<Num> {
    fn div_assign(&mut self, rhs: Num) {
        *self = *self / rhs
    }
}
impl DivAssign<ModInt<Num>> for ModInt<Num> {
    fn div_assign(&mut self, rhs: ModInt<Num>) {
        *self = *self / rhs
    }
}

impl Mul<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;

    fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self * rhs.0
    }
}
impl Mul<Num> for ModInt<Num> {
    type Output = ModInt<Num>;

    fn mul(self, rhs: Num) -> ModInt<Num> {
        let t = (self.0 * rhs) % MOD;
        ModInt(t)
    }
}

impl MulAssign<Num> for ModInt<Num> {
    fn mul_assign(&mut self, rhs: Num) {
        *self = *self * rhs;
    }
}

impl MulAssign<ModInt<Num>> for ModInt<Num> {
    fn mul_assign(&mut self, rhs: ModInt<Num>) {
        *self = *self * rhs;
    }
}

impl ModInt<Num> {
    pub fn pow(self, e: usize) -> ModInt<Num> {
        let mut result = ModInt(1);
        let mut cur = self;
        let mut e = e;
        while e > 0 {
            if e & 1 == 1 {
                result *= cur;
            }
            e >>= 1;
            cur *= cur;
        }
        result
    }
}

pub struct Comb {
    fact: Vec<ModInt<Num>>,
    factinv: Vec<ModInt<Num>>,
}

impl Comb {
    /// Create a object that provides effiecint computation of combinations
    /// for input smaller than `n`.
    ///
    /// This requires `O(n)` time.
    pub fn new(n: usize) -> Comb {
        let mut fact: Vec<ModInt<Num>> = vec![ModInt(0); n + 1];
        let mut factinv: Vec<ModInt<Num>> = vec![ModInt(0); n + 1];
        fact[0] = ModInt(1);
        for i in 0..n {
            fact[i + 1] = fact[i] * (i + 1);
        }
        factinv[n] = fact[n].pow(MOD-2);
        for i in (0..n).rev() {
            factinv[i] = factinv[i + 1] * (i + 1);
        }
        Comb {
            fact: fact,
            factinv: factinv,
        }
    }

    /// `n! = 1 * 2 * ... * n`
    ///
    /// `O(1)` if n is smaller than input in `new` method.
    pub fn fact(&self, n: usize) -> ModInt<Num> {
        if let Some(x) = self.fact.get(n as usize) {
            *x
        } else if n >= MOD {
            ModInt(0)
        } else {
            // Note that this is slow if `n` is large.
            // Precalculation is a possible solution but doesn't work for any module number.
            let mut res = ModInt(1);
            for a in 1..(n + 1) {
                res *= a;
            }
            res
        }
    }

    /// returns `y` such that `n! * y == 1`.
    ///
    /// `O(1)` if n is smaller than input in `new` method.
    pub fn factinv(&self, n: usize) -> ModInt<Num> {
        if let Some(x) = self.factinv.get(n) {
            *x
        } else {
            self.fact(n).pow(MOD-2)
        }
    }

    /// `nPr = n! / (n - r)!`
    ///
    /// `O(1)` if n and r are smaller than input in `new` method.
    pub fn perm(&self, n: usize, r: usize) -> ModInt<Num> {
        if n >= r {
            self.fact(n) * self.factinv(n - r)
        } else {
            ModInt(0)
        }
    }

    /// `nCr = n! / (n - r)! / r!`.
    ///
    /// `O(1)` if n and r are smaller than input in `new` method.
    pub fn comb(&self, n: usize, r: usize) -> ModInt<Num> {
        let m = MOD;
        if n >= m {
            self.comb(n % m, r % m) * self.comb(n / m, r / m) // Lucas' theorem
        } else if n >= r {
            self.fact(n) * self.factinv(n - r) * self.factinv(r)
        } else {
            ModInt(0)
        }
    }
}

fn solve() {
	let (n,k) = read!(usize,usize);
	let a = read!([usize]);
	let mut dp = vec![vec![ModInt(0);n+k+1];k+1];
	dp[0][0] = ModInt(1);
	let mut comb = Comb::new(n);
	for i in 0..k {
		for j in 0..n+1 {
			for k in 0..n+1 {
				if j+k>n {break;}
				if a[i]<k {break;}
				dp[i+1][j+k] += dp[i][j] * comb.comb(n-j,k) * comb.comb(n-k,a[i]-k);
			}
		}
		// for j in 0..n+1 {
		// 	println!("{} {} {}", dp[i+1][j].0, i, j);
		// }
	}
	println!("{}", dp[k][n].0);

}

fn run() {
    solve();
}