Problem:
Score : 100 points

Problem Statement

There are N dishes, numbered 1, 2, \ldots, N.
Initially, for each i (1 \leq i \leq N), Dish i has a_i (1 \leq a_i \leq 3) pieces of sushi on it.

Taro will perform the following operation repeatedly until all the pieces of sushi are eaten:

Roll a die that shows the numbers 1, 2, \ldots, N with equal probabilities, and let i be the outcome. If there are some pieces of sushi on Dish i, eat one of them; if there is none, do nothing.

Find the expected number of times the operation is performed before all the pieces of sushi are eaten.

Constraints

All values in input are integers.

1 \leq N \leq 300

1 \leq a_i \leq 3

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the expected number of times the operation is performed before all the pieces of sushi are eaten.
The output is considered correct when the relative difference is not greater than 10^{-9}.

Sample Input 1

3
1 1 1

Sample Output 1

5.5

The expected number of operations before the first piece of sushi is eaten, is 1.
After that, the expected number of operations before the second sushi is eaten, is 1.5.
After that, the expected number of operations before the third sushi is eaten, is 3.
Thus, the expected total number of operations is 1 + 1.5 + 3 = 5.5.

Sample Input 2

1
3

Sample Output 2

3

Outputs such as 3.00, 3.000000003 and 2.999999997 will also be accepted.

Sample Input 3

2
1 2

Sample Output 3

4.5

Sample Input 4

10
1 3 2 3 3 2 3 2 1 3

Sample Output 4

54.48064457488221

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using pii = pair<int, int>;
using vi = vector<int>;

#ifdef MY_DEBUG_FLAG
template<typename T> void debug(T a) { cerr << a << ' '; }
template<typename T, typename U> void debug(pair<T, U> a) { cerr << a.first << ' ' << a.second << ' '; }
template<typename T> void debug(vector<T> a) { for(auto it : a) debug(it);}
template<typename T> void debug(set<T> a) { for(auto it : a) debug(it);}
#define db(a) cerr << "DEBUG ( " << #a << " == "; debug(a); cerr << ")\n"
#else
#define db(...)
#endif

#define pb push_back
#define eb emplace_back
#define ff first
#define ss second
#define fast ios_base::sync_with_stdio(false), cout.tie(nullptr), cin.tie(nullptr)
#define sz(a) ((int)(a).size())
#define rep(i,a,b) for(int i=(a); i<(b); i++)
#define dec(i,n,a) for(int i=(n); i>=(a); i--)
#define clr(a,v) memset(a, v, sizeof(a))
#define all(a) (a).begin(),(a).end()

constexpr int inf = 0x3f3f3f3f;
constexpr int MAXN = 310;
constexpr int mod = 1000000007;

// p-blty of reaching the state p[a][b][c] with a plates with 1, b w/ 2, c w/ 3
double p[MAXN][MAXN][MAXN], EV[MAXN][MAXN][MAXN];

int main() {
	int n; scanf("%d", &n);
	vi cnt(4);
	rep(i,0,n) {
		int t;
		scanf("%d", &t);
		cnt[t]++;
	}
	p[cnt[1]][cnt[2]][cnt[3]] = 1;

	for(int c = n ; c >= 0; c--) {
		for(int b = n; b >= 0; b--) {
			for(int a = n; a >= 0; a--) {
				if(a+b+c > n || !(a+b+c)) continue;
				double p_waste = (double) (n - a - b - c) / n;
				// pw + pw² + pw³ + ...
				double ev_waste = p_waste / (1 - p_waste);
				EV[a][b][c] += (ev_waste+1)*p[a][b][c];
				if(c) {
					double p_go = (double) c / (a+b+c);
					p[a][b+1][c-1] += p[a][b][c]*p_go;
					EV[a][b+1][c-1] += (EV[a][b][c])*p_go;
				}
				if(b) {
					double p_go = (double) b / (a+b+c);
					p[a+1][b-1][c] += p[a][b][c]*p_go;
					EV[a+1][b-1][c] += (EV[a][b][c])*p_go;
				}
				if(a) {
					double p_go = (double) a / (a+b+c);
					p[a-1][b][c] += p[a][b][c]*p_go;
					EV[a-1][b][c] += (EV[a][b][c])*p_go;
				}
			}
		}
	}
	printf("%.10lf\n", EV[0][0][0]);
}