多重集求组合数,注意到\(n = 20\)所以可以用\(2 ^ n * n\)的容斥来写。
如果没有限制那么答案就是\(C(n + s - 1, n - 1)\)。对每一个限制依次考虑,加上有一种选多的,减去有两种选多的,以此类推。
由于\(n <= 20\),所以组合数事实上是可以\(O(N)\)求的=_=
#include <bits/stdc++.h>
using namespace std;
#define int long long
const int Mod = 1000000007;
const int N = (1 << 20) + 5;
int add (int x, int y) {return (((0ll + (x % Mod) + (y % Mod)) % Mod) + Mod) % Mod;}
int mul (int x, int y) {return (((1ll * (x % Mod) * (y % Mod)) % Mod) + Mod) % Mod;}
int n, s, f[21], fac[21], inv[21];
int C (int n, int m) {
int res = 1;
for (int i = n - m + 1; i <= n; ++i) res = mul (res, i);
res = mul (res, inv[m]);
return res;
}
int fpow (int x, int y) {
int res = 1;
while (y) {
if (y & 1) {
res = mul (res, x);
}
x = mul (x, x);
y >>= 1;
}
return res;
}
signed main () {
cin >> n >> s;
for (int i = 1; i <= n; ++i) {
cin >> f[i];
}
fac[0] = 1;
for (int i = 1; i <= n; ++i) {
fac[i] = mul (fac[i - 1], i);
}
inv[n] = fpow (fac[n], Mod - 2);
for (int i = n; i >= 1; --i) {
inv[i - 1] = mul (inv[i], i);
}
int ans = C (n + s - 1, n - 1);
for (int i = 1; i < (1 << n); ++i) {
int w = 0, del = 0, have_wei = 0;
for (int j = 0; j < n; ++j) {
if (i & (1 << j)) {
have_wei++;
del += f[j + 1] + 1;
}
}
w = have_wei & 1 ? -1 : + 1;
if (s >= del) {
ans = add (ans, w * C (n + s - 1 - del, n - 1));
}
}
cout << ans << endl;
}