Rong Tui [jsoi2011] special products

Description
JYY Led the team to a number of ACM/ICPC match , Brought back a lot of local products , To the students in the lab .
JYY Want to know , Give these specialties to N A classmate , How many different ways are there ？ Of course ,JYY Don't want to be
He Yiyi felt lost because he didn't get the specialty , So every student must get at least one specialty .
for example ,JYY brought 2 Bag fried dough twist and 1 Bag steamed stuffed bun , among A and B Two students , So there is 4 Different species
Distribution method ：
A： Hemp flowers ,B： Hemp flowers 、 Steamed stuffed bun
A： Hemp flowers 、 Hemp flowers ,B： Steamed stuffed bun
A： Steamed stuffed bun ,B： Hemp flowers 、 Hemp flowers
A： Hemp flowers 、 Steamed stuffed bun ,B： Hemp flowers
Input
The first line of input data is the number of students N And the number of specialties M.
The second line contains M It's an integer , Indicates the quantity of each specialty .
N, M No more than 1000, The quantity of each specialty does not exceed 1000
Output
Output one line , Total number of different distribution schemes . Because the output can be huge , You just need to output the final result
MOD 1,000,000,007 Just the value of .
Sample Input
5 4

1 3 3 5
Sample Output
384835

A very good content exclusion question
The original idea was to divide by one arrangement, but in fact, it can't
Find that the specialty itself is independent
So consider it alone
It's no good subtracting one from any score ...
${\prod }_{i=1}^m{C}_{A_i+n-1}^{n-1}$ It's all without consideration
It is actually this thing after tolerance and exclusion
${\sum }_{i=0}^n(-1{)}^i{C}_n^i{\prod }_{j=1}^m{C}_{A_i+n-i-1}^{n-i-1}$

/**************************************************************
    Problem: 4710
    User: Leo_JAM
    Language: C++
    Result: Accepted
    Time:676 ms
    Memory:50360 kb
****************************************************************/
 
#include<iostream>
#include<cstdio>
#include<cstring>
#include<cmath>
#include<algorithm>
using namespace std;
typedef int INT;
#define int long long 
const int mod=1e9+7;
int C[2200][2200];
int F[1200][1200];
void Pre(){
    C[0][0]=1;
    for(int i=1;i<=2000;++i){
        C[i][0]=1;
        for(int j=1;j<=i;++j){
            C[i][j]=(C[i-1][j-1]+C[i-1][j])%mod;
        }
    }
}
int ans=0;
int n,m;
int G[1200];
INT main(){
    Pre();
    scanf("%lld%lld",&n,&m);
    for(int i=1;i<=m;++i)scanf("%lld",&G[i]);
    int cur=-1;
    for(int i=0;i<n;++i){
        cur=cur*(-1);
        int now=cur;
        for(int j=1;j<=m;++j){
            now=(now*C[G[j]+n-i-1][n-i-1]%mod);
        }   
        now=now*C[n][i]%mod;
        ans=(ans+now)%mod;
    }
    cout<<(ans+mod)%mod;
    return 0;
}